Tiger Algebra/core — Japanese @ Weblate: The factorial symbol ! is placed after a positive integer …

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Key FactorialContent

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The factorial symbol ! is placed after a positive integer to indicate that it should be multiplied by every whole number less than it, through <math>1</math>. For example: 
<math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math> 
The factorial operation, denoted as n!, states that: 
<math>n! = n(n–1)(n–2)(n–3) · · · 3 · 2 · 1</math> 
The factorial operation is often used to mathematically express combinations, different ways that selected elements can be arranged when the order does not matter, and permutations, different ways that selected elements can be arranged when the order does matter. 
The factorial of <math>0</math> is <math>1</math>. 
<math>0! = 1</math>

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Key	English	Japanese
DistanceBetweenTwoPointsAndTheirMidpointContent	The distance formula, an application of the Pythagorean theorem, is a very helpful tool for finding the distance between two points. The Pythagorean theorem states the following: in a right triangle, the length of side <math>a</math> squared plus the length of side <math>b</math> squared is equal to the length of the hypotenuse (side <math>c</math>) squared.<br> <math>a^2+b^2=c^2</math><br> <img src="/images/distance-between-two-points-graph1.png" alt="Distance between two points graph"><br><br> The hypotenuse (<math>c</math>) is the longest side of a right triangle and is always opposite the right angle. The length of the hypotenuse also represents the distance between points <b>A</b> and <b>B</b>, which can each be represented by two coordinates: an <math>x</math> coordinate and a <math>y</math> coordinate.<br> Point <b>A</b> = <math>(x_1,y_1)</math><br> Point <b>B</b> = <math>(x_2,y_2)</math><br><br> To get the distance formula, we can rewrite the Pythagorean theorem as:<br> <math>d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}</math><br> in which <math>d</math> represents the distance between points <b>A</b> and <b>B</b>, and the Xs and Ys represent the <math>x</math> and <math>y</math> coordinates of points <b>A</b> and <b>B</b>.<br><br> In order to find the distance between two points, enter their coordinates (for example (1,2) and (3,4)) and click the solve button.	距離公式は、ピタゴラスの定理の応用であり、2点間の距離を見つけるのにとても役立つツールです。ピタゴラスの定理は次のとおりです：直角三角形では、辺<math>a</math>の二乗プラス辺<math>b</math>の二乗は斜辺（辺<math>c</math>）の二乗に等しい。<br><math>a^2+b^2=c^2</math><br><img src="/images/distance-between-two-points-graph1.png" alt="2点間の距離のグラフ"><br><br> 斜辺（<math>c</math>）は直角三角形の最長の辺であり、常に直角の反対側にあります。斜辺の長さはまた、点<b>A</b>と点<b>B</b>の距離を表し、それぞれは2つの座標によって表現できます：<math>x</math>座標と<math>y</math>座標。<br>点<b>A</b> = <math>(x_1,y_1)</math><br>点<b>B</b> = <math>(x_2,y_2)</math><br><br>距離公式を得るために、ピタゴラスの定理を次のように書き換えることができます：<br><math>d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}</math><br>ここで<math>d</math>は点<b>A</b>と点<b>B</b>の距離を表し、XとYは点<b>A</b>と点<b>B</b>の<math>x</math>と<math>y</math>座標を表します。<br><br>2点間の距離を見つけるためには、それぞれの座標（例えば(1,2)と(3,4)）を入力し、ソルブボタンをクリックします。
DividingExponents	Dividing exponents	指数の除算
DividingExponentsContent	<div style="white-space: normal;"> <p>When dividing expressions with exponents, you can use the properties of exponents to simplify the expression. Here's how:</p> <h2>Step 1: Subtract Exponents</h2> <p>To divide two terms with the same base, subtract the exponent of the divisor from the exponent of the dividend. For example, <math>x^a ÷ x^b = x^(a - b)</math>.</p> <p>For instance, if you have <math>x^5 ÷ x^2</math>, you subtract the exponents: <math>x^(5 - 2) = x^3</math>.</p> <h2>Step 2: Simplify</h2> <p>If there are additional terms in the expression, simplify them as well. For example, if you have <math>(x^5 * y^3) ÷ (x^2 * y)</math>, divide the x's and y's separately: <math>x^(5 - 2) * y^(3 - 1) = x^3 * y^2</math>.</p> <p>Remember, when dividing exponents with the same base, subtract the exponents.</p> <p>This process is crucial in algebra and higher-level mathematics when dealing with expressions containing exponents.</p> </div>	タイガー代数ソルバーは、指数を除算する方法を、ステップバイステップで示します。例えば:
DividingFractions	Dividing fractions	分数の除算
DividingFractionsContent	<div style="white-space: normal;"> <p>Dividing fractions is a fundamental operation in mathematics, especially in arithmetic and algebra. When dividing fractions, we follow specific rules to simplify the process and obtain the result.</p> <h2>Basic Concepts</h2> <p>To divide fractions, it's important to understand the following concepts:</p> <ul> <li><strong>Fractions:</strong> Numbers that represent parts of a whole or ratios of two quantities.</li> <li><strong>Dividing fractions:</strong> The process of dividing one fraction by another to find the quotient.</li> <li><strong>Reciprocal:</strong> The reciprocal of a fraction <math>\frac{a}{b}</math> is <math>\frac{b}{a}</math>.</li> </ul> <h2>Division Techniques</h2> <p>There are two common techniques to divide fractions:</p> <ul> <li><strong>Using reciprocals:</strong> Multiply the first fraction by the reciprocal of the second fraction.</li> <li><strong>Converting to multiplication:</strong> Convert the division operation to multiplication by multiplying the first fraction by the reciprocal of the second fraction.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the division of fractions:</p> <h3>Example 1:</h3> <p>Divide <math>\frac{3}{4}</math> by <math>\frac{1}{2}</math></p> <p>Using the reciprocal technique, we multiply the first fraction by the reciprocal of the second fraction:</p> <p><math>\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{6}{4}</math></p> <p>Since <math>\frac{6}{4}</math> can be simplified to <math>\frac{3}{2}</math>, the result is <math>\frac{3}{2}</math>.</p> <h3>Example 2:</h3> <p>Divide <math>\frac{5}{8}</math> by <math>\frac{2}{3}</math></p> <p>We convert the division operation to multiplication by multiplying the first fraction by the reciprocal of the second fraction:</p> <p><math>\frac{5}{8} \div \frac{2}{3} = \frac{5}{8} \times \frac{3}{2}</math></p> <p>After simplification, we get the result as <math>\frac{15}{16}</math>.</p> <h2>Conclusion</h2> <p>Dividing fractions is a fundamental operation that plays a crucial role in various mathematical contexts. Mastering the techniques of fraction division enables efficient problem-solving and enhances mathematical understanding.</p> </div>	タイガー代数計算機は分数を除算し、ステップバイステップの解答を示します。例:
EquationswhichareReducibletoQuadratic	Equations which are reducible to quadratic	2次方程式に還元可能な方程式
EquationswhichareReducibletoQuadraticContent	The Tiger Algebra solver shows you, step by step how to find which equations are reducable to which Quadratic equations. <div style="white-space: normal;"> <p>Some equations can be transformed or manipulated into a quadratic equation, which makes them easier to solve. Equations that can be reduced to quadratic form often involve variables raised to powers or contain multiple terms.</p> <h2>Types of Equations</h2> <p>Equations that are reducible to quadratic form include:</p> <ul> <li><strong>Equations involving radicals:</strong> Equations with square roots or other radicals can often be squared to eliminate the radical and form a quadratic equation.</li> <li><strong>Equations with rational expressions:</strong> Equations containing rational expressions can sometimes be transformed into quadratic equations by clearing denominators.</li> <li><strong>Equations with variable substitutions:</strong> Substituting a new variable can sometimes transform a given equation into a quadratic equation.</li> <li><strong>Equations involving trigonometric functions:</strong> Trigonometric identities or trigonometric substitutions can sometimes reduce trigonometric equations to quadratic form.</li> </ul> <h2>Example</h2> <p>Let's consider the equation <math>\frac{1}{x} - \frac{1}{y} = 3</math>. We can rewrite it as:</p> <div style="margin-left: 20px;"> <math>\frac{1}{x} - \frac{1}{y} = 3 \Rightarrow \frac{y - x}{xy} = 3 \Rightarrow y - x = 3xy</math>.<br> </div> <p>Now, let's make a substitution, <math>z = xy</math>. This transforms the equation into a quadratic equation:</p> <div style="margin-left: 20px;"> <math>y - x = 3xy \Rightarrow y - x = 3z \Rightarrow 3z - y + x = 0</math>.<br> </div> <p>This equation is now in quadratic form, and we can solve it using quadratic equation-solving techniques.</p> <p>Understanding how to manipulate equations into quadratic form can simplify the solving process and make it more manageable.</p> </div>	タイガー代数ソルバーは、どの方程式がどの2次方程式に還元可能かを、ステップバイステップで見つける方法を示します。次の例のようにあなたの方程式を入力してください:
FactoringBinomialsAsSumOrDifferenceOfCubes	Factoring binomials as sum or difference of cubes	立方体の和または差としての二項式の因数分解
FactoringBinomialsAsSumOrDifferenceOfCubesContent	Factoring binomials as the sum or the difference of cubes <br><br> To find the sum or the difference of cubes, you have to apply one of two factoring formulas. They are almost the same, with one small difference: the placement of the minus sign. <br><br> Formula for the sum of cubes: <br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> Formula for the difference of cubes: <br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> In the formula for factoring the sum of cubes, the minus sign is found in the quadratic factor: <math>a^2 – ab + b^2</math>. In the one used to factor the difference of cubes, the minus sign is located in the linear factor: <math>a – b</math>. <br><br> Be sure to apply the appropriate factoring formula! <br><br> Enter the binomial into the calculator and Tiger will show you, step by step, how to solve it.	立方体の和または差としての二項式の因数分解<br><br> 立方体の和または差を見つけるためには、2つの因数分解公式のいずれかを適用する必要があります。ほぼ同じで、マイナス記号の位置だけが異なります。<br><br> 立方体の和の公式:<br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> 立方体の差の公式:<br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> 立方体の和の因数分解公式では、マイナス記号が二次因数 <math>a^2 – ab + b^2</math> に見つけられます。立方体の差を因数分解するための公式では、マイナス記号が線形因数 <math>a – b</math> に位置しています。 <br><br> 適切な因数分解公式を適用しましょう。<br><br> 二項式を計算機に入力し、タイガーがそれを解く方法を、ステップバイステップで示します。
FactoringBinomialsasDifferenceofSquares	Factoring binomials as difference of squares	二項式を完全平方差として因数分解する
FactoringBinomialsasDifferenceofSquaresContent	A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Squares if both terms are perfect squares. Recall we may have to factor out a common factor first. <br><br> If we determine that a binomial is a difference of squares, we factor it into two binomials. The first being the square root of the first term minus the square root of the second term. The second being the square root of the first term plus the square root of the second term. <div style="white-space: normal;"> <p>A binomial is an algebraic expression consisting of two terms. The difference of squares is a special case of factoring where a binomial can be factored into the product of two binomials.</p> <p>The formula for factoring a binomial as the difference of squares is: <math>a^2 - b^2 = (a + b)(a - b)</math>.</p> <h2>Step-by-Step Process</h2> <p>To factor a binomial as the difference of squares, follow these steps:</p> <ol> <li>Identify the square of each term in the binomial.</li> <li>Write the binomial as the difference of squares using the formula above.</li> <li>Factor the expression if possible.</li> </ol> <h2>Example</h2> <p>Let's factor the binomial <math>x^2 - 9</math> as the difference of squares:</p> <p>Step 1: Identify the squares - <math>x^2</math> and <math>9</math> are both perfect squares.</p> <p>Step 2: Write the difference of squares - <math>x^2 - 9 = (x + 3)(x - 3)</math>.</p> <p>Step 3: The expression is now factored as the difference of squares.</p> <p>Factoring binomials as the difference of squares is a fundamental technique in algebra and is often used in various mathematical problems.</p> </div>	二項式が完全平方差として因数分解可能なのは、それが3つのうちどれか一つであることが条件です: 完全平方差、完全立方差、完全立方和。二項式が完全平方差であるとは、両方の項が完全平方であることを指します。まず、共通の因数を取り出す必要があることを覚えておいてください。 <br><br> 二項式が完全平方差であると確認したら、それを二つの二項式に分解します。第一の二項式は、第一項の平方根から第二項の平方根を引いたもので、第二の二項式は、第一項の平方根に第二項の平方根を足したものになります。
FactoringMultiVariablePolynomials	Factoring multivariable polynomials	多変数式を因数分解する
FactoringMultiVariablePolynomialsContent	<div style="white-space: normal;"> <p>Factoring multivariable polynomials involves finding the common factors among the terms of the polynomial and expressing it as the product of simpler polynomials.</p> <h2>Step-by-Step Process</h2> <p>To factor a multivariable polynomial, follow these steps:</p> <ol> <li>Identify common factors among the terms of the polynomial.</li> <li>Factor out the greatest common factor (GCF) from each term.</li> <li>Express the polynomial as the product of the GCF and the remaining factors.</li> <li>If possible, further factor the remaining polynomial using other factoring techniques such as grouping, difference of squares, or trinomial factoring.</li> </ol> <h2>Example</h2> <p>Let's factor the multivariable polynomial <math>2x^2y + 4xy^2</math>:</p> <p>Step 1: Identify common factors - The common factor among the terms is <math>2xy</math>.</p> <p>Step 2: Factor out the GCF - <math>2xy( x + 2y )</math>.</p> <p>Step 3: The polynomial is now factored as <math>2xy( x + 2y )</math>.</p> <p>Factoring multivariable polynomials is essential in algebra and other areas of mathematics for simplifying expressions and solving equations.</p> </div>	タイガー代数計算機は、ステップバイステップで多変数式の因数分解の方法を教えてくれます。
Factorial	Factorials	階乗
FactorialTopic	Factorials	階乗
FactorialContent	The factorial symbol ! is placed after a positive integer to indicate that it should be multiplied by every whole number less than it, through <math>1</math>. For example:<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> The factorial operation, denoted as n!, states that:<br> <math>n! = n(n–1)(n–2)(n–3) · · · 3 · 2 · 1</math><br><br> The factorial operation is often used to mathematically express combinations, different ways that selected elements can be arranged when the order does not matter, and permutations, different ways that selected elements can be arranged when the order does matter.<br><br> The factorial of <math>0</math> is <math>1</math>.<br> <math>0! = 1</math>	階乗記号!は、その正の整数以降の全ての正の整数と乗算を行う必要があることを示します。下に例を示します。<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> ここで、階乗演算はn!と表現され、下記の通りに定義されます。<br> <math>n! = n(n\u20131)(n\u2013 2)(n\u20133) \u00b7 \u00b7 \u00b7 3 \u00b7 2 \u00b7 1</math><br><br> 階乗演算は、組み合わせ（順序を問わない選択要素の異なる配置方法）や置換（順序が重要な選択要素の異なる配置方法）を数学的に表現するためによく使用されます。<br><br> 0の階乗は1です。<br> <math>0! = 1</math>
FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	4項以上の多項式の因数分解
FactoringPolynomialsHavingMoreThan3TermsContent	A simple way to approach factoring a polynomial with four or more terms is by grouping it into sets of two. This method involves examining these sets together to see whether a particular technique could be applied to both. One technique to start with is checking whether it is possible to find the greatest common factor (GCF) between a set of two terms. If the GCF cannot be found, the polynomials could be grouped in another way and examined for another technique. There is always the possibility that the polynomial is prime and cannot be factored. <div style="white-space: normal;"> <p>Factoring polynomials with four or more terms can be more complex than factoring binomials or trinomials. However, there are strategies that can help simplify the process.</p> <h2>Step-by-Step Process</h2> <p>To factor polynomials with four or more terms, follow these steps:</p> <ol> <li>Group the terms in pairs.</li> <li>Factor each pair using common factoring techniques such as GCF, difference of squares, or trinomial factoring.</li> <li>Look for a common factor among the resulting expressions.</li> <li>Factor out the common factor.</li> <li>Express the polynomial as the product of the common factor and the remaining expressions.</li> <li>Check your work by multiplying the factors to ensure you get the original polynomial.</li> </ol> <h2>Example</h2> <p>Let's factor the polynomial <math>x^3 + 2x^2 - 3x - 6</math>:</p> <p>Step 1: Group the terms - <math>(x^3 + 2x^2) - (3x + 6)</math>.</p> <p>Step 2: Factor each pair - <math>x^2(x + 2) - 3(x + 2)</math>.</p> <p>Step 3: Look for a common factor - Both expressions share <math>(x + 2)</math> as a common factor.</p> <p>Step 4: Factor out the common factor - <math>(x + 2)(x^2 - 3)</math>.</p> <p>Step 5: The polynomial is now factored as <math>(x + 2)(x^2 - 3)</math>.</p> <p>Factoring polynomials with four or more terms requires practice and patience, but mastering this skill can greatly simplify algebraic expressions and equations.</p> </div>	4項以上の多項式を因数分解する簡単な方法は、それを2項のセットにグループ分けすることです。この方法では、特定の技術が両方に適用できるかどうかを確認するために、これらのセットを一緒に検討します。開始するための1つのテクニックは、2項のセット間で最大公約数(GCF)を見つけることが可能かどうかを確認することです。GCFが見つからない場合、多項式は別の方法でグループ化でき、別の技術を検討します。常に多項式が素数で、因数分解できない可能性があることを覚えておいてください。
FactorTrinomials	Factor trinomials	3項式の因数分解
FactorTrinomialsContent	<div style="white-space: normal;"> <p>Factoring trinomials is an essential skill in algebra, particularly in polynomial factorization. Trinomials are algebraic expressions with three terms, and factoring involves breaking them down into the product of two or more simpler expressions.</p> <h2>Basic Concepts</h2> <p>To factor trinomials, it's crucial to understand the following concepts:</p> <ul> <li><strong>Trinomials:</strong> Algebraic expressions with three terms, typically in the form <math>ax^2 + bx + c</math>.</li> <li><strong>Factoring:</strong> The process of expressing an algebraic expression as the product of its factors.</li> <li><strong>Factorization techniques:</strong> Methods used to factor trinomials, such as trial and error, grouping, and the quadratic formula.</li> </ul> <h2>Factoring Techniques</h2> <p>There are various techniques to factor trinomials:</p> <ul> <li><strong>Trial and error:</strong> Trying different combinations of factors until finding the correct one.</li> <li><strong>Grouping:</strong> Grouping the terms of the trinomial and factoring common factors.</li> <li><strong>Quadratic formula:</strong> Applying the quadratic formula to find the roots of the trinomial.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the factoring of trinomials:</p> <h3>Example 1:</h3> <p>Factor <math>x^2 + 5x + 6</math></p> <p>We look for two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3. So, <math>x^2 + 5x + 6 = (x + 2)(x + 3)</math></p> <h3>Example 2:</h3> <p>Factor <math>2x^2 + 7x + 3</math></p> <p>Using the quadratic formula, we find the roots to be <math>x = -\frac{1}{2}</math> and <math>x = -3</math>. So, <math>2x^2 + 7x + 3 = 2(x + \frac{1}{2})(x + 3)</math></p> <h2>Conclusion</h2> <p>Factoring trinomials is an important skill in algebra that finds applications in various mathematical problems and real-world scenarios. Mastering the techniques of trinomial factoring enhances problem-solving abilities and deepens understanding of algebraic concepts.</p> </div>	タイガーアルジブラは、異なる方法で3項式を因数分解する方法をステップバイステップで示します。
FindAreaofTrianglegivenbyits3sides	Find area of triangle given by its 3 sides	その3辺によって与えられた三角形の面積を求める
FindAreaofTrianglegivenbyits3sidesContent	To find the Area of Triangle given its 3 sides, enter the sizes of the three sides seperated by commas and wrapped in parentheses. Tiger Algebra will show you the step by step solution. Sample input: <math>(3,4,5)</math> <div style="white-space: normal;"> <p>To find the area of a triangle given its three sides (also known as the side-length formula), you can use Heron's formula. Heron's formula allows you to calculate the area of a triangle using its side lengths alone.</p> <h2>Heron's Formula</h2> <p>Heron's formula states that the area <math>A</math> of a triangle with side lengths <math>a</math>, <math>b</math>, and <math>c</math> is given by:</p> <div style="margin-left: 20px;"> <math>A = \sqrt{s(s - a)(s - b)(s - c)}</math>,<br> <i>where</i> <math>s</math> is the semi-perimeter of the triangle, calculated as:<br> <math>s = \frac{a + b + c}{2}</math>. </div> <h2>Example</h2> <p>Let's find the area of a triangle with side lengths 5, 6, and 7:</p> <div style="margin-left: 20px;"> <math>a = 5, b = 6, c = 7</math><br> <math>s = \frac{5 + 6 + 7}{2} = 9</math><br> <math>A = \sqrt{9(9 - 5)(9 - 6)(9 - 7)}</math><br> <math>A = \sqrt{9(4)(3)(2)} = \sqrt{216}</math><br> <math>A \approx 14.7</math>. </div> <p>Therefore, the area of the triangle is approximately 14.7 square units.</p> <p>Heron's formula provides an efficient method for finding the area of a triangle when you know the lengths of its sides.</p> </div>	3辺の大きさが与えられた三角形の面積を求めるには、3つの辺のサイズをカンマで区切り、括弧で囲んで入力します。タイガーアルジブラはステップバイステップの解を表示します。サンプル入力: <math>(3,4,5)</math>
FindPrimeFactors	Find prime factors	素因数を見つける
FindPrimeFactorsContent	<div style="white-space: normal;"> <p>Prime factorization is the process of finding the prime numbers that multiply together to give a specific number. This is often used in mathematics to simplify fractions or solve problems involving factors.</p> <h2>Step 1: Begin with the Number</h2> <p>To find the prime factors of a number, start with the number itself. Let's take an example: 72.</p> <h2>Step 2: Divide by Prime Numbers</h2> <p>Start dividing the number by the smallest prime number, 2, and continue dividing by prime numbers until you cannot divide evenly anymore.</p> <p>For example, for 72:</p> <ul> <li>72 ÷ 2 = 36</li> <li>36 ÷ 2 = 18</li> <li>18 ÷ 2 = 9</li> <li>9 ÷ 3 = 3</li> <li>3 is already a prime number.</li> </ul> <h2>Step 3: Record the Prime Factors</h2> <p>The prime factors of 72 are 2, 2, 2, 3, and 3.</p> <h2>Step 4: Check Your Work</h2> <p>Multiply all the prime factors together to ensure you get the original number: 2 * 2 * 2 * 3 * 3 = 72.</p> <p>Finding prime factors is essential for many mathematical operations, including simplifying fractions, finding greatest common divisors, and more.</p> </div>	素数は1または自分自身でしか割り切ることができません。合成数（非素数）の整数はその素因数の積として記述できます。因子（任意の数）または素数（任意の数）をアルジブラ計算機に入力すると、タイガーはその素因数を見つけてくれます。
GCFCalculator	Greatest common factor	最大公約数
GCFCalculatorContent	The greatest common factor (GCF), sometimes also referred to as the highest common factor (HCF) or the greatest common divisor (GCD), is the largest positive integer that a set of integers can all be divided by. For example, the largest number that 12, 24, and 32 can all be divided by is 4, so their greatest common factor is 4. Similarly, the largest number that 3, 5, and 10 can all be divided by is 1, so their greatest common factor is 1. <br><br> There are two ways to find the greatest common factor: listing the factors of each number and prime factorization. <br><br> <b>Method 1: Listing the factors of each number</b><br> List all of the factors of each number and identify the largest (greatest) factor that all the numbers have in common. <br><br> 12 - 1, 2, 3, <b>4</b>, 6, 12<br> 24 - 1, 2, 3, <b>4</b>, 6, 8, 12, 24<br> 32 - 1, 2, 3, <b>4</b>, 8, 16, 32<br><br> 4 is the greatest factor the numbers have in common, so it is the greatest common factor. <br><br> <b>Method 2: Prime factorization</b><br> Use a factor tree to identify the prime factors (factors that are prime numbers) of each number. Identify the prime factors that all the numbers have in common and multiply them together to get the greatest common factor. <br> <img src="/images/gcfTree1.png" alt="Prime factorization tree"><br> The prime factors that all of the numbers have in common are 2 and 2. Multiply these together to get the greatest common factor, 4.	最大公約数（GCF）は、ときどき最高公約数（HCF）または最大公約数（GCD）とも呼ばれ、一組の整数すべてが分割できる最大の正の整数です。たとえば、12、24、32すべてが分割できる最大の数は4なので、その最大公約数は4です。同様に、3、5、10すべてが分割できる最大の数は1なので、その最大公約数は1です。 <br><br> 最大公約数を見つける方法は2つあります：それぞれの数字の因数をリストアップする方法と、素因数分解する方法です。 <br><br> <b>方法1：それぞれの数の因数をリストアップする</b><br> それぞれの数のすべての因数をリストアップし、すべての数が共通に持っている最大（最大）の因数を特定します。 <br><br> 12 - 1, 2, 3, <b>4</b>, 6, 12<br> 24 - 1, 2, 3, <b>4</b>, 6, 8, 12, 24<br> 32 - 1, 2, 3, <b>4</b>, 8, 16, 32<br><br> 4は、数が共通に持つ最大の因数なので、最大公約数です。 <br><br> <b>方法2：素因数分解</b><br> 因数のツリーを使用して、それぞれの数の素因数（素数である因数）を特定します。すべての数が共通に持っている素因数を特定し、それらを乗算して最大公約数を取得します。 <br> <img src="/images/gcfTree1.png" alt="素因数分解のツリー"><br> すべての数が共通に持つ素因数は2と2です。これらを乗算して最大公約数、4を得ます。
GeometricSeries	Geometric series	幾何級数
GeometricSeriesContent	In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series <math>1,2,4,8</math> is geometric, because each successive term can be obtained by multiplying the previous term by <math>2</math>. <div style="white-space: normal;"> <p>A geometric series is the sum of the terms of a geometric sequence. It is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.</p> <h2>General Form</h2> <p>The general form of a geometric series is:</p> <div style="margin-left: 20px;"> <math>a + ar + ar^2 + ar^3 + \ldots = \sum_{n=0}^{\infty} ar^n</math>, </div> <p>where <math>a</math> is the first term and <math>r</math> is the common ratio.</p> <h2>Sum Formula</h2> <p>The sum of a finite geometric series with <math>n</math> terms is given by the formula:</p> <div style="margin-left: 20px;"> <math>S_n = a \frac{1 - r^n}{1 - r}</math>, </div> <p>where <math>S_n</math> is the sum of the first <math>n</math> terms.</p> <h2>Properties</h2> <ul> <li>If the absolute value of the common ratio <math>r</math> is less than 1, the series converges to a finite value.</li> <li>If the absolute value of <math>r</math> is greater than or equal to 1, the series diverges.</li> <li>The sum of an infinite geometric series can be found using the formula for the sum of an infinite geometric series:</li> <div style="margin-left: 20px;"> <math>S_{\infty} = \frac{a}{1 - r}</math>. </div> </ul> <h2>Applications</h2> <p>Geometric series have various applications in mathematics, physics, engineering, and finance. They are used to model growth and decay processes, calculate interest, analyze circuits, and more.</p> <p>Understanding geometric series and their properties is essential for solving problems in many fields.</p> </div>	数学において、幾何級数とは連続する項と項の間の比率が一定である級数のことを言います。例えば、級数 <math>1,2,4,8</math> は幾何級数であり、それは各項が前の項を <math>2</math> 倍にすることで得られるからです。
IdentifyingPerfectCubes	Identifying perfect cubes	完全立方体の識別
IdentifyingPerfectCubesContent	<div style="white-space: normal;"> <p>In general, a perfect cube is represented as <math>n^3</math>, where <math>n</math> is an integer.</p> <p>There are several techniques to identify perfect cubes:</p> <ul> <li><strong>Examine the last digit:</strong> Perfect cubes have specific patterns in their last digit. By examining the last digit of a number, you can often determine if it's a perfect cube.</li> <li><strong>Prime factorization:</strong> Another method is to perform prime factorization of the number and check if each prime factor appears in groups of three.</li> <li><strong>Use a table of cubes:</strong> Memorizing the cubes of numbers from 1 to 10 can help quickly identify perfect cubes.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the identification of perfect cubes:</p> <h3>Example 1:</h3> <p>Identify whether 64 is a perfect cube.</p> <p>Using the examination of the last digit method, we find that the last digit of 64 is 4. The last digits of perfect cubes cycle in a specific pattern (0, 1, 8, 7, 4, 5, 6, 3, 2, 9), and 4 is in this cycle. Hence, 64 is a perfect cube.</p> <h3>Example 2:</h3> <p>Identify whether 1000 is a perfect cube.</p> <p>By performing prime factorization, we find that <math>1000 = 2^3 \times 5^3</math>. Since both 2 and 5 appear in groups of three, 1000 is a perfect cube.</p> <h2>Conclusion</h2> <p>Identifying perfect cubes is an important skill in mathematics that finds applications in various fields. By understanding the basic concepts and techniques, one can easily recognize perfect cubes and utilize them in problem-solving.</p> </div>	完全立方体は別の多項式の立方体です。Tiger Algebra Solverは完全立方体を見つけ出し、その手順を一つずつ教えてくれます。
LCMCalculator	Least common multiple (LCM)	最小公倍数（LCM）

Key	English	Japanese
ExponentialEquationsUsingLogarithms.plugin	solving exponential equations using logarithms	対数を使用して指数方程式を解く
ExponentialEquationsUsingLogarithmsResultTypeDecimal	Decimal form:	十進数:
ExponentialEquationsUsingLogarithmsStep1NTCommonLog	Take the common logarithm of both sides of the equation:	方程式の両側に共通対数を取ります：
ExponentialEquationsUsingLogarithmsStep1NTLogRule	Use the log rule: <span class="formula-look no-text"><math>log_a(x^y)=y*log_a(x)</math></span> to move the exponent outside the logarithm:	対数のルール：<span class="formula-look no-text"><math>log_a(x^y)=y*log_a(x)</math></span> を使用して指数を対数の外に移動します：
ExponentialEquationsUsingLogarithmsStep1TextUnit1	<math>{expression}={integer}</math><br><br>Take the common logarithm of both sides of the equation:<br><br> <math>log_10({expression})=log_10({integer})</math><br><br> Use the log rule: <span class="formula-look no-text"><math>log_a(x^y)=</math><math>ylog_a(x)</math></span> to move the exponent outside the logarithm:<br><br> <math>{variable}log_10({base})=log_10({integer})</math>	<math>{expression}={integer}</math><br><br>方程式の両側に共通対数を取ります：<br><br> <math>log_10({expression})=log_10({integer})</math><br><br> 対数のルール： <span class="formula-look no-text"><math>log_a(x^y)=</math><math>ylog_a(x)</math></span> を使用して指数を対数の外に移動します：<br><br> <math>{variable}log_10({base})=log_10({integer})</math>
ExponentialEquationsUsingLogarithmsStep1Title1	Remove the variable from the exponent using logarithms	対数を使用して指数から変数を取り除きます
ExponentialEquationsUsingLogarithmsStep2NTDecimalForm	Decimal form:	十進数:
ExponentialEquationsUsingLogarithmsStep2NTDivide	Divide both sides of the equation by <math>log_10({base})</math>:	方程式の両側を <math>log_10({base})</math> で割ります：
ExponentialEquationsUsingLogarithmsStep2NTUseFormula	Use the formula <span class="formula-look no-text"><math>\frac {log_b(x)}{log_b(a)}=log_a(x)</math></span> to combine the logarithms into one:	対数を一つにまとめるための式 <span class="formula-look no-text"><math>\frac {log_b(x)}{log_b(a)}=log_a(x)</math></span> を使用します：
ExponentialEquationsUsingLogarithmsStep2TextUnit1	<math>{var}*log_10({base})=log_({integer})</math><br><br> Divide the equation by <math>log_10({base})</math>:<br><br> <math>{var}=log_10({integer})/log_10({base})</math><br>	<math>{var}*log_10({base})=log_({integer})</math><br><br> 方程式を <math>log_10({base})</math> で割ります：<br><br> <math>{var}=log_10({integer})/log_10({base})</math><br>
ExponentialEquationsUsingLogarithmsStep2Title1	Isolate the {var}-variable	{var}-変数を単独にします
ExponentialEquationsUsingLogarithmsTopicContent	An exponential equation is an equation with a variable exponent or an exponent with a variable in it. For example: <math>2^x=256</math> and <math>3^(2x-4)=342</math> are both exponential equations. <br>We can solve exponential equations in one of two ways, depending on the bases of the equation's terms. <br><br><b>Solving exponential equations using logarithms</b><br> The first way to solve exponential equations does not take the bases into account and involves using the following logarithmic rule to move and isolate the equation's variable: <br><br> <math>log_x a^b=b*log_x a</math> <br><br> Finding the log of a number with a variable as an exponent allows us to move the exponent to the front of the equation, making it a multiplier on the log. From there, we can isolate the variable and solve the equation. <br><br> <a href="/en/solution/exponential-equations-using-logarithms/2%5Ex=256/">See an example problem here</a> <br><br> <b>Solving exponential equations using exponent properties</b><br> The second way to solve exponential equations uses properties of exponents. If we can get both sides of the equation to have the same base, then we can set the exponents equal to each other. This relationship can be expressed as:<br> if <math>x^a=x^b</math> then <math>a=b</math> <br><br>for example:<br> <math>2^x=256</math> <br> Because <math>256=2^8</math> then <math>2^x=2^8</math>, meaning <math>x=8</math>. </p>	指数方程式とは、変数指数またはそれに変数が含まれる指数を持つ方程式のことを指します。例えば、<math>2^x=256</math>と<math>3^(2x-4)=342</math>は指数方程式です。 <br>指数方程式の解き方は、その基数によって２通りあります。 <br><br><b>対数を用いて指数方程式を解く</b><br> 指数方程式を解く最初の方法は、基数を考慮にいれず、対数ルールを用いて方程式の変数を移動・分離します。 <br><br> <math>log_x a^b=b*log_x a</math> <br><br> 指数が変数の数の対数を求めると、指数を先頭に移動できます、これにより対数の前に掛ける数として使うことができます。そこから変数を単独で求め方程式の解を求めることができます。 <br><br> <a href="/en/solution/exponential-equations-using-logarithms/2%5Ex=256/">ここに例題があります</a> <br><br> <b>指数の性質を用いて指数方程式を解く</b><br> 指数方程式を解く2つ目の方法は指数の性質を用います。もし方程式の両辺が同じ基数であれば、その指数を等しいと設定できます。この関係性は次の通り表せます： if <math>x^a=x^b</math> then <math>a=b</math> <br><br>例えば：<br> <math>2^x=256</math> <br> <math>256=2^8</math>なので <math>2^x=2^8</math>、ということは<math>x=8</math>です。 </p>
ExponentialEquationsUsingLogarithmsTopicTitle	Exponential equations	指数方程式
Exponents	Exponents	指数
Factorial	Factorials	階乗
FactorialContent	The factorial symbol ! is placed after a positive integer to indicate that it should be multiplied by every whole number less than it, through <math>1</math>. For example:<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> The factorial operation, denoted as n!, states that:<br> <math>n! = n(n–1)(n–2)(n–3) · · · 3 · 2 · 1</math><br><br> The factorial operation is often used to mathematically express combinations, different ways that selected elements can be arranged when the order does not matter, and permutations, different ways that selected elements can be arranged when the order does matter.<br><br> The factorial of <math>0</math> is <math>1</math>.<br> <math>0! = 1</math>	階乗記号!は、その正の整数以降の全ての正の整数と乗算を行う必要があることを示します。下に例を示します。<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> ここで、階乗演算はn!と表現され、下記の通りに定義されます。<br> <math>n! = n(n\u20131)(n\u2013 2)(n\u20133) \u00b7 \u00b7 \u00b7 3 \u00b7 2 \u00b7 1</math><br><br> 階乗演算は、組み合わせ（順序を問わない選択要素の異なる配置方法）や置換（順序が重要な選択要素の異なる配置方法）を数学的に表現するためによく使用されます。<br><br> 0の階乗は1です。<br> <math>0! = 1</math>
Factorial.plugin	factorials	factorials
FactorialStep1TextUnit1	The factorial of {factorial} is the product of all positive integers less than or equal to {factorial}:<br><br> <math>{solution}</math>	{factorial}の階乗は、{factorial}以下のすべての正の整数の積です:<br><br> <math>{solution}</math>
FactorialStep1Title1	Find the factorial	階乗を求める
FactorialTopic	Factorials	階乗
FactoringBinomialsasDifferenceofSquares	Factoring binomials as difference of squares	二項式を完全平方差として因数分解する
FactoringBinomialsasDifferenceofSquaresContent	A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Squares if both terms are perfect squares. Recall we may have to factor out a common factor first. <br><br> If we determine that a binomial is a difference of squares, we factor it into two binomials. The first being the square root of the first term minus the square root of the second term. The second being the square root of the first term plus the square root of the second term. <div style="white-space: normal;"> <p>A binomial is an algebraic expression consisting of two terms. The difference of squares is a special case of factoring where a binomial can be factored into the product of two binomials.</p> <p>The formula for factoring a binomial as the difference of squares is: <math>a^2 - b^2 = (a + b)(a - b)</math>.</p> <h2>Step-by-Step Process</h2> <p>To factor a binomial as the difference of squares, follow these steps:</p> <ol> <li>Identify the square of each term in the binomial.</li> <li>Write the binomial as the difference of squares using the formula above.</li> <li>Factor the expression if possible.</li> </ol> <h2>Example</h2> <p>Let's factor the binomial <math>x^2 - 9</math> as the difference of squares:</p> <p>Step 1: Identify the squares - <math>x^2</math> and <math>9</math> are both perfect squares.</p> <p>Step 2: Write the difference of squares - <math>x^2 - 9 = (x + 3)(x - 3)</math>.</p> <p>Step 3: The expression is now factored as the difference of squares.</p> <p>Factoring binomials as the difference of squares is a fundamental technique in algebra and is often used in various mathematical problems.</p> </div>	二項式が完全平方差として因数分解可能なのは、それが3つのうちどれか一つであることが条件です: 完全平方差、完全立方差、完全立方和。二項式が完全平方差であるとは、両方の項が完全平方であることを指します。まず、共通の因数を取り出す必要があることを覚えておいてください。 <br><br> 二項式が完全平方差であると確認したら、それを二つの二項式に分解します。第一の二項式は、第一項の平方根から第二項の平方根を引いたもので、第二の二項式は、第一項の平方根に第二項の平方根を足したものになります。
FactoringBinomialsAsSumOrDifferenceOfCubes	Factoring binomials as sum or difference of cubes	立方体の和または差としての二項式の因数分解
FactoringBinomialsAsSumOrDifferenceOfCubesContent	Factoring binomials as the sum or the difference of cubes <br><br> To find the sum or the difference of cubes, you have to apply one of two factoring formulas. They are almost the same, with one small difference: the placement of the minus sign. <br><br> Formula for the sum of cubes: <br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> Formula for the difference of cubes: <br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> In the formula for factoring the sum of cubes, the minus sign is found in the quadratic factor: <math>a^2 – ab + b^2</math>. In the one used to factor the difference of cubes, the minus sign is located in the linear factor: <math>a – b</math>. <br><br> Be sure to apply the appropriate factoring formula! <br><br> Enter the binomial into the calculator and Tiger will show you, step by step, how to solve it.	立方体の和または差としての二項式の因数分解<br><br> 立方体の和または差を見つけるためには、2つの因数分解公式のいずれかを適用する必要があります。ほぼ同じで、マイナス記号の位置だけが異なります。<br><br> 立方体の和の公式:<br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> 立方体の差の公式:<br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> 立方体の和の因数分解公式では、マイナス記号が二次因数 <math>a^2 – ab + b^2</math> に見つけられます。立方体の差を因数分解するための公式では、マイナス記号が線形因数 <math>a – b</math> に位置しています。 <br><br> 適切な因数分解公式を適用しましょう。<br><br> 二項式を計算機に入力し、タイガーがそれを解く方法を、ステップバイステップで示します。
FactoringMultiVariablePolynomials	Factoring multivariable polynomials	多変数式を因数分解する
FactoringMultiVariablePolynomialsContent	<div style="white-space: normal;"> <p>Factoring multivariable polynomials involves finding the common factors among the terms of the polynomial and expressing it as the product of simpler polynomials.</p> <h2>Step-by-Step Process</h2> <p>To factor a multivariable polynomial, follow these steps:</p> <ol> <li>Identify common factors among the terms of the polynomial.</li> <li>Factor out the greatest common factor (GCF) from each term.</li> <li>Express the polynomial as the product of the GCF and the remaining factors.</li> <li>If possible, further factor the remaining polynomial using other factoring techniques such as grouping, difference of squares, or trinomial factoring.</li> </ol> <h2>Example</h2> <p>Let's factor the multivariable polynomial <math>2x^2y + 4xy^2</math>:</p> <p>Step 1: Identify common factors - The common factor among the terms is <math>2xy</math>.</p> <p>Step 2: Factor out the GCF - <math>2xy( x + 2y )</math>.</p> <p>Step 3: The polynomial is now factored as <math>2xy( x + 2y )</math>.</p> <p>Factoring multivariable polynomials is essential in algebra and other areas of mathematics for simplifying expressions and solving equations.</p> </div>	タイガー代数計算機は、ステップバイステップで多変数式の因数分解の方法を教えてくれます。
FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	4項以上の多項式の因数分解
FactoringPolynomialsHavingMoreThan3TermsContent	A simple way to approach factoring a polynomial with four or more terms is by grouping it into sets of two. This method involves examining these sets together to see whether a particular technique could be applied to both. One technique to start with is checking whether it is possible to find the greatest common factor (GCF) between a set of two terms. If the GCF cannot be found, the polynomials could be grouped in another way and examined for another technique. There is always the possibility that the polynomial is prime and cannot be factored. <div style="white-space: normal;"> <p>Factoring polynomials with four or more terms can be more complex than factoring binomials or trinomials. However, there are strategies that can help simplify the process.</p> <h2>Step-by-Step Process</h2> <p>To factor polynomials with four or more terms, follow these steps:</p> <ol> <li>Group the terms in pairs.</li> <li>Factor each pair using common factoring techniques such as GCF, difference of squares, or trinomial factoring.</li> <li>Look for a common factor among the resulting expressions.</li> <li>Factor out the common factor.</li> <li>Express the polynomial as the product of the common factor and the remaining expressions.</li> <li>Check your work by multiplying the factors to ensure you get the original polynomial.</li> </ol> <h2>Example</h2> <p>Let's factor the polynomial <math>x^3 + 2x^2 - 3x - 6</math>:</p> <p>Step 1: Group the terms - <math>(x^3 + 2x^2) - (3x + 6)</math>.</p> <p>Step 2: Factor each pair - <math>x^2(x + 2) - 3(x + 2)</math>.</p> <p>Step 3: Look for a common factor - Both expressions share <math>(x + 2)</math> as a common factor.</p> <p>Step 4: Factor out the common factor - <math>(x + 2)(x^2 - 3)</math>.</p> <p>Step 5: The polynomial is now factored as <math>(x + 2)(x^2 - 3)</math>.</p> <p>Factoring polynomials with four or more terms requires practice and patience, but mastering this skill can greatly simplify algebraic expressions and equations.</p> </div>	4項以上の多項式を因数分解する簡単な方法は、それを2項のセットにグループ分けすることです。この方法では、特定の技術が両方に適用できるかどうかを確認するために、これらのセットを一緒に検討します。開始するための1つのテクニックは、2項のセット間で最大公約数(GCF)を見つけることが可能かどうかを確認することです。GCFが見つからない場合、多項式は別の方法でグループ化でき、別の技術を検討します。常に多項式が素数で、因数分解できない可能性があることを覚えておいてください。
Factorization	Factorization	因数分解
FactorTrinomials	Factor trinomials	3項式の因数分解

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階乗記号!は、その正の整数以降の全ての正の整数と乗算を行う必要があることを示します。下に例を示します。 
<math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math> 
ここで、階乗演算はn!と表現され、下記の通りに定義されます。 
<math>n! = n(n\u20131)(n\u2013 2)(n\u20133) \u00b7 \u00b7 \u00b7 3 \u00b7 2 \u00b7 1</math> 
階乗演算は、組み合わせ（順序を問わない選択要素の異なる配置方法）や置換（順序が重要な選択要素の異なる配置方法）を数学的に表現するためによく使用されます。 
0の階乗は1です。 
<math>0! = 1</math>

9 hours ago

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FactorialContent

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9 hours ago

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9 hours ago

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core/TigerAlgebra.Localization/Resources/TigerSharedResource.ja.resx, string 462