Precalculus Examples

$\frac{\sqrt[3]{x^{8} y^{2}}}{\sqrt{x y}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{8} y^{2}$ as ${(x^{2})}^{3} (x^{2} y^{2})$ .

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Step 1.1.1

Factor out $x^{6}$ .

$\frac{\sqrt[3]{x^{6} x^{2} y^{2}}}{\sqrt{x y}}$

Step 1.1.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}}}{\sqrt{x y}}$

Step 2

Multiply $\frac{x^{2} \sqrt[3]{x^{2} y^{2}}}{\sqrt{x y}}$ by $\frac{\sqrt{x y}}{\sqrt{x y}}$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}}}{\sqrt{x y}} \cdot \frac{\sqrt{x y}}{\sqrt{x y}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{x^{2} \sqrt[3]{x^{2} y^{2}}}{\sqrt{x y}}$ by $\frac{\sqrt{x y}}{\sqrt{x y}}$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{\sqrt{x y} \sqrt{x y}}$

Step 3.1.2

Raise $\sqrt{x y}$ to the power of $1$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{\sqrt{x y}}^{1} \sqrt{x y}}$

Step 3.1.3

Raise $\sqrt{x y}$ to the power of $1$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{\sqrt{x y}}^{1} {\sqrt{x y}}^{1}}$

Step 3.1.4

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

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{\sqrt{x y}}^{1 + 1}}$

Step 3.1.5

Add $1$ and $1$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{\sqrt{x y}}^{2}}$

Step 3.1.6

Rewrite ${\sqrt{x y}}^{2}$ as $x y$ .

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Step 3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{({(x y)}^{\frac{1}{2}})}^{2}}$

Step 3.1.6.2

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

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{(x y)}^{\frac{1}{2} \cdot 2}}$

Step 3.1.6.3

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

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

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

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

Step 3.1.6.4.2

Rewrite the expression.

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{{(x y)}^{1}}$

Step 3.1.6.5

Simplify.

$\frac{x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{x y}$

Step 3.2

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

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Step 3.2.1

Factor $x$ out of $x^{2} \sqrt[3]{x^{2} y^{2}} \sqrt{x y}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $x$ out of $x y$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

$\frac{x \sqrt[3]{x^{2} y^{2}} \sqrt{x y}}{y}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

$\frac{x ({(x y)}^{\frac{1}{2}} \sqrt[3]{x^{2} y^{2}})}{y}$

Step 4.1.2

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{3}{6}}$ .

$\frac{x ({(x y)}^{\frac{3}{6}} \sqrt[3]{x^{2} y^{2}})}{y}$

Step 4.1.3

Rewrite ${(x y)}^{\frac{3}{6}}$ as $\sqrt[6]{{(x y)}^{3}}$ .

$\frac{x (\sqrt[6]{{(x y)}^{3}} \sqrt[3]{x^{2} y^{2}})}{y}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} y^{2}}$ as ${(x^{2} y^{2})}^{\frac{1}{3}}$ .

$\frac{x (\sqrt[6]{{(x y)}^{3}} {(x^{2} y^{2})}^{\frac{1}{3}})}{y}$

Step 4.1.5

Rewrite ${(x^{2} y^{2})}^{\frac{1}{3}}$ as ${(x^{2} y^{2})}^{\frac{2}{6}}$ .

$\frac{x (\sqrt[6]{{(x y)}^{3}} {(x^{2} y^{2})}^{\frac{2}{6}})}{y}$

Step 4.1.6

Rewrite ${(x^{2} y^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(x^{2} y^{2})}^{2}}$ .

$\frac{x (\sqrt[6]{{(x y)}^{3}} \sqrt[6]{{(x^{2} y^{2})}^{2}})}{y}$

Step 4.2

Combine using the product rule for radicals.

$\frac{x \sqrt[6]{{(x y)}^{3} {(x^{2} y^{2})}^{2}}}{y}$

Step 4.3

Apply the product rule to $x y$ .

$\frac{x \sqrt[6]{x^{3} y^{3} {(x^{2} y^{2})}^{2}}}{y}$

Step 4.4

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

$\frac{x \sqrt[6]{x^{3} y^{3} {(x^{2})}^{2} {(y^{2})}^{2}}}{y}$

Step 4.5

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

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Step 4.5.1

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

$\frac{x \sqrt[6]{x^{3} y^{3} x^{2 \cdot 2} {(y^{2})}^{2}}}{y}$

Step 4.5.2

Multiply $2$ by $2$ .

$\frac{x \sqrt[6]{x^{3} y^{3} x^{4} {(y^{2})}^{2}}}{y}$

Step 4.6

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

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Step 4.6.1

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

$\frac{x \sqrt[6]{x^{3} y^{3} x^{4} y^{2 \cdot 2}}}{y}$

Step 4.6.2

Multiply $2$ by $2$ .

$\frac{x \sqrt[6]{x^{3} y^{3} x^{4} y^{4}}}{y}$

Step 4.7

Combine exponents.

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Step 4.7.1

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 4.7.1.1

Move $x^{4}$ .

$\frac{x \sqrt[6]{x^{4} x^{3} y^{3} y^{4}}}{y}$

Step 4.7.1.2

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

$\frac{x \sqrt[6]{x^{4 + 3} y^{3} y^{4}}}{y}$

Step 4.7.1.3

Add $4$ and $3$ .

$\frac{x \sqrt[6]{x^{7} y^{3} y^{4}}}{y}$

Step 4.7.2

Multiply $y^{3}$ by $y^{4}$ by adding the exponents.

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Step 4.7.2.1

Move $y^{4}$ .

$\frac{x \sqrt[6]{x^{7} (y^{4} y^{3})}}{y}$

Step 4.7.2.2

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

$\frac{x \sqrt[6]{x^{7} y^{4 + 3}}}{y}$

Step 4.7.2.3

Add $4$ and $3$ .

$\frac{x \sqrt[6]{x^{7} y^{7}}}{y}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $x^{7} y^{7}$ as ${(x y)}^{6} (x y)$ .

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Step 5.1.1

Factor out $x^{6}$ .

$\frac{x \sqrt[6]{x^{6} x y^{7}}}{y}$

Step 5.1.2

Factor out $y^{6}$ .

$\frac{x \sqrt[6]{x^{6} x (y^{6} y)}}{y}$

Step 5.1.3

Move $x$ .

$\frac{x \sqrt[6]{x^{6} y^{6} x y}}{y}$

Step 5.1.4

Rewrite $x^{6} y^{6}$ as ${(x y)}^{6}$ .

$\frac{x \sqrt[6]{{(x y)}^{6} x y}}{y}$

Step 5.1.5

Add parentheses.

$\frac{x \sqrt[6]{{(x y)}^{6} (x y)}}{y}$

Step 5.2

Pull terms out from under the radical.

$\frac{x \cdot x y \sqrt[6]{x y}}{y}$

Step 5.3

Combine exponents.

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Step 5.3.1

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

$\frac{x^{1} x y \sqrt[6]{x y}}{y}$

Step 5.3.2

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

$\frac{x^{1} x^{1} y \sqrt[6]{x y}}{y}$

Step 5.3.3

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

$\frac{x^{1 + 1} y \sqrt[6]{x y}}{y}$

Step 5.3.4

Add $1$ and $1$ .

$\frac{x^{2} y \sqrt[6]{x y}}{y}$

Step 6

Cancel the common factor of $y$ .

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Step 6.1

Cancel the common factor.

$\frac{x^{2} y \sqrt[6]{x y}}{y}$

Step 6.2

Divide $x^{2} \sqrt[6]{x y}$ by $1$ .

$x^{2} \sqrt[6]{x y}$