Algebra Examples

$\frac{{(8 x^{2} y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Apply the product rule to $8 x^{2} y^{5}$ .

$\frac{{(8 x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.1.2

Apply the product rule to $8 x^{2}$ .

$\frac{8^{\frac{1}{3}} {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\frac{{(2^{3})}^{\frac{1}{3}} {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.3

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

$\frac{2^{3 (\frac{1}{3})} {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.4

Cancel the common factor of $3$ .

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Step 1.4.1

Cancel the common factor.

$\frac{2^{3 (\frac{1}{3})} {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.4.2

Rewrite the expression.

$\frac{2^{1} {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.5

Evaluate the exponent.

$\frac{2 {(x^{2})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.6

Multiply the exponents in ${(x^{2})}^{\frac{1}{3}}$ .

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Step 1.6.1

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

$\frac{2 x^{2 (\frac{1}{3})} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.6.2

Combine $2$ and $\frac{1}{3}$ .

$\frac{2 x^{\frac{2}{3}} {(y^{5})}^{\frac{1}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 1.7

Multiply the exponents in ${(y^{5})}^{\frac{1}{3}}$ .

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Step 1.7.1

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

$\frac{2 x^{\frac{2}{3}} y^{5 (\frac{1}{3})}}{{(x y)}^{\frac{2}{3}}}$

Step 1.7.2

Combine $5$ and $\frac{1}{3}$ .

$\frac{2 x^{\frac{2}{3}} y^{\frac{5}{3}}}{{(x y)}^{\frac{2}{3}}}$

Step 2

Simplify the expression.

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Step 2.1

Apply the product rule to $x y$ .

$\frac{2 x^{\frac{2}{3}} y^{\frac{5}{3}}}{x^{\frac{2}{3}} y^{\frac{2}{3}}}$

Step 2.2

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

$\frac{2 x^{\frac{2}{3}} y^{\frac{5}{3}} y^{- \frac{2}{3}}}{x^{\frac{2}{3}}}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $y^{\frac{5}{3}}$ by $y^{- \frac{2}{3}}$ by adding the exponents.

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Step 3.1.1

Move $y^{- \frac{2}{3}}$ .

$\frac{2 x^{\frac{2}{3}} (y^{- \frac{2}{3}} y^{\frac{5}{3}})}{x^{\frac{2}{3}}}$

Step 3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{\frac{2}{3}} y^{- \frac{2}{3} + \frac{5}{3}}}{x^{\frac{2}{3}}}$

Step 3.1.3

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{2}{3}} y^{\frac{- 2 + 5}{3}}}{x^{\frac{2}{3}}}$

Step 3.1.4

Add $- 2$ and $5$ .

$\frac{2 x^{\frac{2}{3}} y^{\frac{3}{3}}}{x^{\frac{2}{3}}}$

Step 3.1.5

Divide $3$ by $3$ .

$\frac{2 x^{\frac{2}{3}} y^{1}}{x^{\frac{2}{3}}}$

Step 3.2

Simplify $2 x^{\frac{2}{3}} y^{1}$ .

$\frac{2 x^{\frac{2}{3}} y}{x^{\frac{2}{3}}}$

Step 4

Cancel the common factor.

$\frac{2 x^{\frac{2}{3}} y}{x^{\frac{2}{3}}}$

Step 5

Divide $2 y$ by $1$ .

$2 y$