Algebra Examples

$\frac{(\frac{3}{4}) (x^{- \frac{1}{2}})}{y^{- \frac{2}{3}}} = \frac{a}{b}$

Step 1

Simplify both sides.

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Step 1.1

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{3}{4}}{y^{- \frac{2}{3}} x^{\frac{1}{2}}} = \frac{a}{b}$

Step 1.2

Move $y^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{\frac{3}{4} y^{\frac{2}{3}}}{x^{\frac{1}{2}}} = \frac{a}{b}$

Step 1.3

Combine $\frac{3}{4}$ and $y^{\frac{2}{3}}$ .

$\frac{\frac{3 y^{\frac{2}{3}}}{4}}{x^{\frac{1}{2}}} = \frac{a}{b}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 y^{\frac{2}{3}}}{4} \cdot \frac{1}{x^{\frac{1}{2}}} = \frac{a}{b}$

Step 1.5

Combine.

$\frac{3 y^{\frac{2}{3}} \cdot 1}{4 x^{\frac{1}{2}}} = \frac{a}{b}$

Step 1.6

Multiply $3$ by $1$ .

$\frac{3 y^{\frac{2}{3}}}{4 x^{\frac{1}{2}}} = \frac{a}{b}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$3 y^{\frac{2}{3}} b = 4 x^{\frac{1}{2}} a$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $4 x^{\frac{1}{2}} a = 3 y^{\frac{2}{3}} b$ .

$4 x^{\frac{1}{2}} a = 3 y^{\frac{2}{3}} b$

Step 3.2

Divide each term in $4 x^{\frac{1}{2}} a = 3 y^{\frac{2}{3}} b$ by $4 a$ and simplify.

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Step 3.2.1

Divide each term in $4 x^{\frac{1}{2}} a = 3 y^{\frac{2}{3}} b$ by $4 a$ .

$\frac{4 x^{\frac{1}{2}} a}{4 a} = \frac{3 y^{\frac{2}{3}} b}{4 a}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

$\frac{4 x^{\frac{1}{2}} a}{4 a} = \frac{3 y^{\frac{2}{3}} b}{4 a}$

Step 3.2.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} a}{a} = \frac{3 y^{\frac{2}{3}} b}{4 a}$

Step 3.2.2.3

Cancel the common factor.

$\frac{x^{\frac{1}{2}} a}{a} = \frac{3 y^{\frac{2}{3}} b}{4 a}$

Step 3.2.2.4

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{3 y^{\frac{2}{3}} b}{4 a}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Reorder factors in $\frac{3 y^{\frac{2}{3}} b}{4 a}$ .

$x^{\frac{1}{2}} = \frac{3 b y^{\frac{2}{3}}}{4 a}$

Step 3.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = {(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.4.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.4.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = {(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$

Step 3.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$

Step 3.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$

Step 3.4.1.1.2

Simplify.

$x = {(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(\frac{3 b y^{\frac{2}{3}}}{4 a})}^{2}$ .

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Step 3.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.2.1.1.1

Apply the product rule to $\frac{3 b y^{\frac{2}{3}}}{4 a}$ .

$x = \frac{{(3 b y^{\frac{2}{3}})}^{2}}{{(4 a)}^{2}}$

Step 3.4.2.1.1.2

Apply the product rule to $3 b y^{\frac{2}{3}}$ .

$x = \frac{{(3 b)}^{2} {(y^{\frac{2}{3}})}^{2}}{{(4 a)}^{2}}$

Step 3.4.2.1.1.3

Apply the product rule to $3 b$ .

$x = \frac{3^{2} b^{2} {(y^{\frac{2}{3}})}^{2}}{{(4 a)}^{2}}$

Step 3.4.2.1.1.4

Apply the product rule to $4 a$ .

$x = \frac{3^{2} b^{2} {(y^{\frac{2}{3}})}^{2}}{4^{2} a^{2}}$

Step 3.4.2.1.2

Simplify the numerator.

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Step 3.4.2.1.2.1

Raise $3$ to the power of $2$ .

$x = \frac{9 b^{2} {(y^{\frac{2}{3}})}^{2}}{4^{2} a^{2}}$

Step 3.4.2.1.2.2

Multiply the exponents in ${(y^{\frac{2}{3}})}^{2}$ .

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Step 3.4.2.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{9 b^{2} y^{\frac{2}{3} \cdot 2}}{4^{2} a^{2}}$

Step 3.4.2.1.2.2.2

Multiply $\frac{2}{3} \cdot 2$ .

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Step 3.4.2.1.2.2.2.1

Combine $\frac{2}{3}$ and $2$ .

$x = \frac{9 b^{2} y^{\frac{2 \cdot 2}{3}}}{4^{2} a^{2}}$

Step 3.4.2.1.2.2.2.2

Multiply $2$ by $2$ .

$x = \frac{9 b^{2} y^{\frac{4}{3}}}{4^{2} a^{2}}$

Step 3.4.2.1.3

Raise $4$ to the power of $2$ .

$x = \frac{9 b^{2} y^{\frac{4}{3}}}{16 a^{2}}$