Example Question - algebra
Here are examples of questions we've helped users solve.
Understanding Exponential Expressions
<p>To simplify the expression, we use the property of exponents that states:</p> <p>When dividing like bases, you subtract the exponents:</p> <p>\(\frac{x^a}{x^b} = x^{a-b}\)</p> <p>Thus, the equation holds:</p> <p>\(\frac{x^a}{x^b} = x^{a-b}\)</p>
Linear Equation in Two Variables
<p>To solve the equation \(3x - 5y - 21 = 0\), we can isolate \(y\).</p> <p>Add \(5y\) and \(21\) to both sides:</p> <p>\(3x = 5y + 21\)</p> <p>Now, subtract \(21\) from both sides:</p> <p>\(3x - 21 = 5y\)</p> <p>Divide each term by \(5\) to solve for \(y\):</p> <p>\(y = \frac{3x - 21}{5}\)</p>
Simplifying an Algebraic Expression
<p> Primero, simplificamos la expresión dentro de los corchetes: </p> <p> \[ \left( \frac{2}{7} \right)^2 \cdot \left( -\frac{1}{6} \right)^2 = \frac{4}{49} \cdot \frac{1}{36} \] </p> <p> Multiplicamos las fracciones: </p> <p> \[ \frac{4 \cdot 1}{49 \cdot 36} = \frac{4}{1764} \] </p> <p> Finalmente, simplificamos \(\frac{4}{1764}\): </p> <p> \[ \frac{4 \div 4}{1764 \div 4} = \frac{1}{441} \] </p> <p> Por lo tanto, el resultado final es: </p> <p> \[ \left( \left( \frac{2}{7} \right)^2 \cdot \left( -\frac{1}{6} \right)^2 \right)^2 = \left( \frac{1}{441} \right)^2 = \frac{1}{194481} \] </p>
Simplifying Algebraic Expressions
<p>Para simplificar la expresión dada:</p> <p>Comenzamos con:</p> <p> \(\frac{x^2 \cdot (y^5)^3 \cdot \left( \frac{1}{z} \right)^2}{\frac{x^3}{z^5} \cdot \left( \frac{y^2}{z} \right)^{-4}}\)</p> <p>Reescribiendo y simplificando paso a paso:</p> <p>Numerador:</p> <p> \(x^2 \cdot y^{15} \cdot \frac{1}{z^2}\)</p> <p>Denominador:</p> <p> \(\frac{x^3}{z^5} \cdot \frac{z^4}{y^8}\)</p> <p>Combinando:</p> <p> \(\frac{x^2 \cdot y^{15} \cdot z^4}{x^3 \cdot y^8 \cdot z^2 \cdot z^5}\)</p> <p>Reduciendo términos:</p> <p> \(= \frac{y^{15 - 8}}{z^{2 + 5 - 4} \cdot x^{3 - 2}}\)</p> <p>Resultado final:</p> <p> \(\frac{y^7}{z^5 \cdot x}\)</p>
Simplifying Square Roots
<p>Para simplificar la expresión, empezamos con la fracción: \(\frac{\sqrt{5}}{1 + \sqrt{5}}\).</p> <p>Multiplicamos el numerador y el denominador por el conjugado del denominador: \(\frac{\sqrt{5}(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})}\).</p> <p>Esto da como resultado: \(\frac{\sqrt{5} - 5}{1 - 5} = \frac{\sqrt{5} - 5}{-4} = -\frac{\sqrt{5}}{4} + \frac{5}{4}\).</p> <p>La simplificación final da como resultado: \(\sqrt{5}\).</p>
Simplifying Algebraic Expressions
<p>For the first expression, \(7h + 21\), you can factor out the common factor:</p> <p>Factorization yields \(7(h + 3)\).</p> <p>For the second expression, \(6x - 12\), you can also factor out the common factor:</p> <p>Factorization yields \(6(x - 2)\).</p>
Expression Simplification
<p>To simplify the expression, we first expand the terms:</p> <p>1. 5x(2a) + 5x(3) - 4(6a) = 10ax + 15x - 24a.</p> <p>2. Combine like terms with 10x² + 15d - 8x - 12.</p> <p>Final expression: 10x² + (15d - 8x) - 24a - 12</p>
Polynomial Expression Simplification
<p>First, group like terms in the expression.</p> <p>5u(2u) + 5u(3) - 4c => 10u^2 + 15u - 4c.</p> <p>Combine this with the second part of the expression:</p> <p>10u^2 + 15u - 8u - 12.</p> <p>This simplifies to:</p> <p>10u^2 + (15u - 8u) - 12 = 10u^2 + 7u - 12.</p> <p>The final simplified expression is:</p> <p>10u^2 + 7u - 12.</p>
Algebraic Expression Evaluation
<p>For the expressions given:</p> <p>1. \(4(y-2) = 4y - 8\)</p> <p>2. \(5f(2f+7) = 10f^2 + 35f\)</p> <p>3. \((5x-4)(2x+3) = 10x^2 + 15x - 8x - 12 = 10x^2 + 7x - 12\)</p>
Algebraic Expressions Evaluation
<p>To evaluate the expressions:</p> <p>1. \( 4(y-2) = 4y - 8 \)</p> <p>2. \( 5f(2f+7) = 10f^2 + 35f \)</p> <p>3. \( (5x-4)(2x+3) = 10x^2 + 15x - 8 \)</p>
Algebraic Expression Expansion
<p>1. Expand \(3(4-x)\):</p> <p> \(= 12 - 3x\)</p> <p>2. Expand \(4(y-2)\):</p> <p> \(= 4y - 8\)</p> <p>3. Expand \(5f(2f+7)\):</p> <p> \(= 10f^2 + 35f\)</p>
Finding the Value of n
<p>Given \( (x^n)^3 = \frac{x^{18}}{x^{-6}} \), we can start by simplifying the right side:</p> <p>First, rewrite \( x^{-6} \) as \( \frac{1}{x^6} \), so we have:</p> <p>\( \frac{x^{18}}{x^{-6}} = x^{18} \cdot x^{6} = x^{18 + 6} = x^{24} \)</p> <p>Now we have:</p> <p> \( (x^n)^3 = x^{24} \)</p> <p>Using the property of exponents, we get:</p> <p> \( x^{3n} = x^{24} \)</p> <p>Since the bases are the same, set the exponents equal:</p> <p> \( 3n = 24 \)</p> <p>Now, solving for \( n \):</p> <p> \( n = \frac{24}{3} = 8 \)</p> <p>Thus, the value of \( n \) is \( 8 \).</p>
Simplifying an Expression Involving Exponents
<p>Given the expression:</p> <p>\(\frac{10^{-a}}{7^5 \times 10^7 \times 7^{-7}}\)</p> <p>First, simplify the denominator:</p> <p>\(7^5 \times 7^{-7} = 7^{5 - 7} = 7^{-2}\)</p> <p>Now, rewrite the entire expression:</p> <p>\(\frac{10^{-a}}{7^{-2} \times 10^7}\)</p> <p>This can be rewritten as:</p> <p> \(\frac{10^{-a}}{10^7} \times 7^{2}\)</p> <p>Now, simplify the powers of 10:</p> <p> \(10^{-a - 7} \times 7^2\)</p> <p>Thus, the final simplified expression is:</p> <p>\(7^2 \times 10^{-(a + 7)}\)</p> ```
Simplifying a Mathematical Expression
<p>Start with the expression:</p> <p>\(\frac{15^{16}}{15^{4} \times (15^{2})^{3}}\)</p> <p>First, simplify \((15^{2})^{3}\):</p> <p>\((15^{2})^{3} = 15^{6}\)</p> <p>Now substitute back into the expression:</p> <p>\(\frac{15^{16}}{15^{4} \times 15^{6}}\)</p> <p>Combine the terms in the denominator:</p> <p>So, the denominator becomes \(15^{4 + 6} = 15^{10}\)</p> <p>Now the expression is:</p> <p>\(\frac{15^{16}}{15^{10}}\)</p> <p>Using the quotient rule of exponents:</p> <p>Subtract the exponents: \(15^{16 - 10} = 15^{6}\)</p> <p>Thus, the simplified expression is:</p> <p>\(15^{6}\)</p>
Finding the Value of a Variable in an Equation
<p>Given the equation \( x^{2m} = \frac{(x^3)^8}{x^6} \).</p> <p>First, simplify the right side:</p> <p>\( \frac{(x^3)^8}{x^6} = \frac{x^{24}}{x^6} = x^{24-6} = x^{18} \).</p> <p>Now, equate the exponents:</p> <p>So, \( 2m = 18 \).</p> <p>To find \( m \), divide both sides by 2:</p> <p>Therefore, \( m = \frac{18}{2} = 9 \).</p>
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