Algebra Examples

$\frac{5}{\sqrt[4]{8 x^{3} y^{3}}}$

Step 1

Multiply $\frac{5}{\sqrt[4]{8 x^{3} y^{3}}}$ by $\frac{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}$ .

$\frac{5}{\sqrt[4]{8 x^{3} y^{3}}} \cdot \frac{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{5}{\sqrt[4]{8 x^{3} y^{3}}}$ by $\frac{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{3}}$ .

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{\sqrt[4]{8 x^{3} y^{3}} {\sqrt[4]{8 x^{3} y^{3}}}^{3}}$

Step 2.2

Raise $\sqrt[4]{8 x^{3} y^{3}}$ to the power of $1$ .

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{1} {\sqrt[4]{8 x^{3} y^{3}}}^{3}}$

Step 2.3

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

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{1 + 3}}$

Step 2.4

Add $1$ and $3$ .

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{\sqrt[4]{8 x^{3} y^{3}}}^{4}}$

Step 2.5

Rewrite ${\sqrt[4]{8 x^{3} y^{3}}}^{4}$ as $8 x^{3} y^{3}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8 x^{3} y^{3}}$ as ${(8 x^{3} y^{3})}^{\frac{1}{4}}$ .

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{({(8 x^{3} y^{3})}^{\frac{1}{4}})}^{4}}$

Step 2.5.2

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

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{(8 x^{3} y^{3})}^{\frac{1}{4} \cdot 4}}$

Step 2.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{(8 x^{3} y^{3})}^{\frac{4}{4}}}$

Step 2.5.4

Cancel the common factor of $4$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{(8 x^{3} y^{3})}^{\frac{4}{4}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{{(8 x^{3} y^{3})}^{1}}$

Step 2.5.5

Simplify.

$\frac{5 {\sqrt[4]{8 x^{3} y^{3}}}^{3}}{8 x^{3} y^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[4]{8 x^{3} y^{3}}}^{3}$ as $\sqrt[4]{{(8 x^{3} y^{3})}^{3}}$ .

$\frac{5 \sqrt[4]{{(8 x^{3} y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.2

Apply the product rule to $8 x^{3} y^{3}$ .

$\frac{5 \sqrt[4]{{(8 x^{3})}^{3} {(y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.3

Apply the product rule to $8 x^{3}$ .

$\frac{5 \sqrt[4]{8^{3} {(x^{3})}^{3} {(y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.4

Raise $8$ to the power of $3$ .

$\frac{5 \sqrt[4]{512 {(x^{3})}^{3} {(y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.5

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 3.5.1

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

$\frac{5 \sqrt[4]{512 x^{3 \cdot 3} {(y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.5.2

Multiply $3$ by $3$ .

$\frac{5 \sqrt[4]{512 x^{9} {(y^{3})}^{3}}}{8 x^{3} y^{3}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{5 \sqrt[4]{512 x^{9} y^{3 \cdot 3}}}{8 x^{3} y^{3}}$

Step 3.6.2

Multiply $3$ by $3$ .

$\frac{5 \sqrt[4]{512 x^{9} y^{9}}}{8 x^{3} y^{3}}$

Step 3.7

Rewrite $512 x^{9} y^{9}$ as ${(4 x^{2} y^{2})}^{4} \cdot (2 x y)$ .

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Step 3.7.1

Factor $256$ out of $512$ .

$\frac{5 \sqrt[4]{256 (2) x^{9} y^{9}}}{8 x^{3} y^{3}}$

Step 3.7.2

Rewrite $256$ as $4^{4}$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 x^{9} y^{9}}}{8 x^{3} y^{3}}$

Step 3.7.3

Factor out $x^{8}$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 (x^{8} x) y^{9}}}{8 x^{3} y^{3}}$

Step 3.7.4

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 ({(x^{2})}^{4} x) y^{9}}}{8 x^{3} y^{3}}$

Step 3.7.5

Factor out $y^{8}$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 ({(x^{2})}^{4} x) (y^{8} y)}}{8 x^{3} y^{3}}$

Step 3.7.6

Rewrite $y^{8}$ as ${(y^{2})}^{4}$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 ({(x^{2})}^{4} x) ({(y^{2})}^{4} y)}}{8 x^{3} y^{3}}$

Step 3.7.7

Move $x$ .

$\frac{5 \sqrt[4]{4^{4} \cdot 2 ({(x^{2})}^{4}) {(y^{2})}^{4} x y}}{8 x^{3} y^{3}}$

Step 3.7.8

Move $2$ .

$\frac{5 \sqrt[4]{4^{4} ({(x^{2})}^{4}) {(y^{2})}^{4} \cdot 2 x y}}{8 x^{3} y^{3}}$

Step 3.7.9

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

$\frac{5 \sqrt[4]{{(4 x^{2} y^{2})}^{4} \cdot 2 x y}}{8 x^{3} y^{3}}$

Step 3.7.10

Add parentheses.

$\frac{5 \sqrt[4]{{(4 x^{2} y^{2})}^{4} \cdot 2 (x y)}}{8 x^{3} y^{3}}$

Step 3.7.11

Add parentheses.

$\frac{5 \sqrt[4]{{(4 x^{2} y^{2})}^{4} \cdot (2 x y)}}{8 x^{3} y^{3}}$

Step 3.8

Pull terms out from under the radical.

$\frac{5 \cdot 4 x^{2} y^{2} \sqrt[4]{2 x y}}{8 x^{3} y^{3}}$

Step 3.9

Multiply $5$ by $4$ .

$\frac{20 x^{2} y^{2} \sqrt[4]{2 x y}}{8 x^{3} y^{3}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $20$ and $8$ .

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Step 4.1.1

Factor $4$ out of $20 x^{2} y^{2} \sqrt[4]{2 x y}$ .

$\frac{4 (5 x^{2} y^{2} \sqrt[4]{2 x y})}{8 x^{3} y^{3}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $4$ out of $8 x^{3} y^{3}$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 4.2.1

Factor $x^{2}$ out of $5 x^{2} y^{2} \sqrt[4]{2 x y}$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $y^{2}$ and $y^{3}$ .

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Step 4.3.1

Factor $y^{2}$ out of $5 y^{2} \sqrt[4]{2 x y}$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

$\frac{5 \sqrt[4]{2 x y}}{2 x y}$