Pre-Algebra Examples

$\frac{{(x^{8} \cdot y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $x^{8} y^{6}$ .

$\frac{{(x^{8})}^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.2

Multiply the exponents in ${(x^{8})}^{\frac{1}{2}}$ .

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Step 1.2.1

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

$\frac{x^{8 (\frac{1}{2})} {(y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $8$ .

$\frac{x^{2 (4) \frac{1}{2}} {(y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.2.2.2

Cancel the common factor.

$\frac{x^{2 \cdot 4 \frac{1}{2}} {(y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.2.2.3

Rewrite the expression.

$\frac{x^{4} {(y^{6})}^{\frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.3

Multiply the exponents in ${(y^{6})}^{\frac{1}{2}}$ .

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Step 1.3.1

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

$\frac{x^{4} y^{6 (\frac{1}{2})}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Factor $2$ out of $6$ .

$\frac{x^{4} y^{2 (3) \frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.3.2.2

Cancel the common factor.

$\frac{x^{4} y^{2 \cdot 3 \frac{1}{2}}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 1.3.2.3

Rewrite the expression.

$\frac{x^{4} y^{3}}{{(x^{6} \cdot y^{- 3})}^{\frac{1}{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x^{4} y^{3}}{{(x^{6} \cdot \frac{1}{y^{3}})}^{\frac{1}{3}}}$

Step 2.2

Combine $x^{6}$ and $\frac{1}{y^{3}}$ .

$\frac{x^{4} y^{3}}{{(\frac{x^{6}}{y^{3}})}^{\frac{1}{3}}}$

Step 2.3

Apply the product rule to $\frac{x^{6}}{y^{3}}$ .

$\frac{x^{4} y^{3}}{\frac{{(x^{6})}^{\frac{1}{3}}}{{(y^{3})}^{\frac{1}{3}}}}$

Step 2.4

Multiply the exponents in ${(x^{6})}^{\frac{1}{3}}$ .

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Step 2.4.1

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

$\frac{x^{4} y^{3}}{\frac{x^{6 (\frac{1}{3})}}{{(y^{3})}^{\frac{1}{3}}}}$

Step 2.4.2

Cancel the common factor of $3$ .

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Step 2.4.2.1

Factor $3$ out of $6$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 2.5

Simplify the denominator.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

Cancel the common factor of $3$ .

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Step 2.5.1.2.1

Cancel the common factor.

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

Step 2.5.1.2.2

Rewrite the expression.

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

Step 2.5.2

Simplify.

$\frac{x^{4} y^{3}}{\frac{x^{2}}{y}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$x^{4} y^{3} \frac{y}{x^{2}}$

Step 4

Cancel the common factor of $x^{2}$ .

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Step 4.1

Factor $x^{2}$ out of $x^{4} y^{3}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Raise $y$ to the power of $1$ .

$x^{2} (y^{1} y^{3})$

Step 6

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

$x^{2} y^{1 + 3}$

Step 7

Add $1$ and $3$ .

$x^{2} y^{4}$