Precalculus Examples

$\frac{\sqrt{32 x^{7} y^{5}}}{\sqrt[3]{2 x y}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $32 x^{7} y^{5}$ as ${(4 x^{3} y^{2})}^{2} \cdot (2 x y)$ .

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Step 1.1.1

Factor $16$ out of $32$ .

$\frac{\sqrt{16 (2) x^{7} y^{5}}}{\sqrt[3]{2 x y}}$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 2 x^{7} y^{5}}}{\sqrt[3]{2 x y}}$

Step 1.1.3

Factor out $x^{6}$ .

$\frac{\sqrt{4^{2} \cdot 2 (x^{6} x) y^{5}}}{\sqrt[3]{2 x y}}$

Step 1.1.4

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

$\frac{\sqrt{4^{2} \cdot 2 ({(x^{3})}^{2} x) y^{5}}}{\sqrt[3]{2 x y}}$

Step 1.1.5

Factor out $y^{4}$ .

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

Step 1.1.6

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Step 1.1.7

Move $x$ .

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

Step 1.1.8

Move $2$ .

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

Step 1.1.9

Rewrite $(4^{2} ({(x^{3})}^{2})) {(y^{2})}^{2}$ as ${(4 x^{3} y^{2})}^{2}$ .

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

Step 1.1.10

Add parentheses.

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

Step 1.1.11

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{4 x^{3} y^{2} \sqrt{2 x y}}{\sqrt[3]{2 x y}}$ by $\frac{{\sqrt[3]{2 x y}}^{2}}{{\sqrt[3]{2 x y}}^{2}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{4 x^{3} y^{2} \sqrt{2 x y}}{\sqrt[3]{2 x y}}$ by $\frac{{\sqrt[3]{2 x y}}^{2}}{{\sqrt[3]{2 x y}}^{2}}$ .

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

Step 3.2

Raise $\sqrt[3]{2 x y}$ to the power of $1$ .

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

Step 3.3

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

$\frac{4 x^{3} y^{2} \sqrt{2 x y} {\sqrt[3]{2 x y}}^{2}}{{\sqrt[3]{2 x y}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

Rewrite ${\sqrt[3]{2 x y}}^{3}$ as $2 x y$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Cancel the common factor of $4$ and $2$ .

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Step 4.1

Factor $2$ out of $4 x^{3} y^{2} \sqrt{2 x y} {\sqrt[3]{2 x y}}^{2}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $2 x y$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $x$ out of $x y$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Factor $y$ out of $y^{1}$ .

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.2.5

Divide $2 x^{2} y \sqrt{2 x y} {\sqrt[3]{2 x y}}^{2}$ by $1$ .

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

Step 7

Rewrite ${\sqrt[3]{2 x y}}^{2}$ as $\sqrt[3]{{(2 x y)}^{2}}$ .

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

Step 8

Apply the product rule to $2 x y$ .

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

Step 9

Apply the product rule to $2 x$ .

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

Step 10

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

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

Step 11

Multiply $2 x^{2} y \sqrt{2 x y} \sqrt[3]{4 x^{2} y^{2}}$ .

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Step 11.1

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

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Step 11.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2} y^{2}}$ as ${(4 x^{2} y^{2})}^{\frac{1}{3}}$ .

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

Step 11.1.2

Rewrite ${(4 x^{2} y^{2})}^{\frac{1}{3}}$ as ${(4 x^{2} y^{2})}^{\frac{2}{6}}$ .

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

Step 11.1.3

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

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

Step 11.1.4

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

$2 x^{2} y (\sqrt[6]{{(4 x^{2} y^{2})}^{2}} {(2 x y)}^{\frac{1}{2}})$

Step 11.1.5

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

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

Step 11.1.6

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

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

Step 11.2

Combine using the product rule for radicals.

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

Step 11.3

Apply the product rule to $4 x^{2} y^{2}$ .

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

Step 11.4

Apply the product rule to $2 x y$ .

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

Step 11.5

Apply the product rule to $4 x^{2}$ .

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

Step 11.6

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

$2 x^{2} y \sqrt[6]{16 {(x^{2})}^{2} {(y^{2})}^{2} {(2 x)}^{3} y^{3}}$

Step 11.7

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

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Step 11.7.1

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

$2 x^{2} y \sqrt[6]{16 x^{2 \cdot 2} {(y^{2})}^{2} {(2 x)}^{3} y^{3}}$

Step 11.7.2

Multiply $2$ by $2$ .

$2 x^{2} y \sqrt[6]{16 x^{4} {(y^{2})}^{2} {(2 x)}^{3} y^{3}}$

Step 11.8

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

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Step 11.8.1

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

$2 x^{2} y \sqrt[6]{16 x^{4} y^{2 \cdot 2} {(2 x)}^{3} y^{3}}$

Step 11.8.2

Multiply $2$ by $2$ .

$2 x^{2} y \sqrt[6]{16 x^{4} y^{4} {(2 x)}^{3} y^{3}}$

Step 11.9

Apply the product rule to $2 x$ .

$2 x^{2} y \sqrt[6]{16 x^{4} y^{4} (2^{3} x^{3}) y^{3}}$

Step 11.10

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

$2 x^{2} y \sqrt[6]{16 x^{4} y^{4} \cdot 8 x^{3} y^{3}}$

Step 11.11

Combine exponents.

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Step 11.11.1

Multiply $8$ by $16$ .

$2 x^{2} y \sqrt[6]{128 x^{4} y^{4} x^{3} y^{3}}$

Step 11.11.2

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

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Step 11.11.2.1

Move $x^{3}$ .

$2 x^{2} y \sqrt[6]{128 (x^{3} x^{4}) y^{4} y^{3}}$

Step 11.11.2.2

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

$2 x^{2} y \sqrt[6]{128 x^{3 + 4} y^{4} y^{3}}$

Step 11.11.2.3

Add $3$ and $4$ .

$2 x^{2} y \sqrt[6]{128 x^{7} y^{4} y^{3}}$

Step 11.11.3

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

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Step 11.11.3.1

Move $y^{3}$ .

$2 x^{2} y \sqrt[6]{128 x^{7} (y^{3} y^{4})}$

Step 11.11.3.2

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

$2 x^{2} y \sqrt[6]{128 x^{7} y^{3 + 4}}$

Step 11.11.3.3

Add $3$ and $4$ .

$2 x^{2} y \sqrt[6]{128 x^{7} y^{7}}$

Step 12

Rewrite $128 x^{7} y^{7}$ as ${(2 x y)}^{6} \cdot (2 x y)$ .

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Step 12.1

Factor $64$ out of $128$ .

$2 x^{2} y \sqrt[6]{64 (2) x^{7} y^{7}}$

Step 12.2

Rewrite $64$ as $2^{6}$ .

$2 x^{2} y \sqrt[6]{2^{6} \cdot 2 x^{7} y^{7}}$

Step 12.3

Factor out $x^{6}$ .

$2 x^{2} y \sqrt[6]{2^{6} \cdot 2 (x^{6} x) y^{7}}$

Step 12.4

Factor out $y^{6}$ .

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

Step 12.5

Move $x$ .

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

Step 12.6

Move $2$ .

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

Step 12.7

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

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

Step 12.8

Add parentheses.

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

Step 12.9

Add parentheses.

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

Step 13

Pull terms out from under the radical.

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

Step 14

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 14.1

Move $x$ .

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

Step 14.2

Multiply $x$ by $x^{2}$ .

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Step 14.2.1

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

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

Step 14.2.2

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

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

Step 14.3

Add $1$ and $2$ .

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

Step 15

Multiply $y$ by $y$ by adding the exponents.

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Step 15.1

Move $y$ .

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

Step 15.2

Multiply $y$ by $y$ .

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

Step 16

Multiply $2$ by $2$ .

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