Algebra Examples

$\sqrt[4]{\frac{3 a^{2}}{8 b^{7}}}$

Step 1

Rewrite $\frac{3 a^{2}}{8 b^{7}}$ as ${(\frac{1}{b})}^{4} \frac{3 a^{2}}{8 b^{3}}$ .

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Step 1.1

Factor the perfect power $1^{4}$ out of $3 a^{2}$ .

$\sqrt[4]{\frac{1^{4} (3 a^{2})}{8 b^{7}}}$

Step 1.2

Factor the perfect power $b^{4}$ out of $8 b^{7}$ .

$\sqrt[4]{\frac{1^{4} (3 a^{2})}{b^{4} \cdot (8 b^{3})}}$

Step 1.3

Rearrange the fraction $\frac{1^{4} (3 a^{2})}{b^{4} \cdot (8 b^{3})}$ .

$\sqrt[4]{{(\frac{1}{b})}^{4} \frac{3 a^{2}}{8 b^{3}}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{b} \sqrt[4]{\frac{3 a^{2}}{8 b^{3}}}$

Step 3

Rewrite $\sqrt[4]{\frac{3 a^{2}}{8 b^{3}}}$ as $\frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}}$ .

$\frac{1}{b} \cdot \frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}}$

Step 4

Combine.

$\frac{1 \sqrt[4]{3 a^{2}}}{b \sqrt[4]{8 b^{3}}}$

Step 5

Multiply $\sqrt[4]{3 a^{2}}$ by $1$ .

$\frac{\sqrt[4]{3 a^{2}}}{b \sqrt[4]{8 b^{3}}}$

Step 6

Multiply $\frac{\sqrt[4]{3 a^{2}}}{b \sqrt[4]{8 b^{3}}}$ by $\frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$ .

$\frac{\sqrt[4]{3 a^{2}}}{b \sqrt[4]{8 b^{3}}} \cdot \frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt[4]{3 a^{2}}}{b \sqrt[4]{8 b^{3}}}$ by $\frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b \sqrt[4]{8 b^{3}} {\sqrt[4]{8 b^{3}}}^{3}}$

Step 7.2

Move $\sqrt[4]{8 b^{3}}$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b (\sqrt[4]{8 b^{3}} {\sqrt[4]{8 b^{3}}}^{3})}$

Step 7.3

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b ({\sqrt[4]{8 b^{3}}}^{1} {\sqrt[4]{8 b^{3}}}^{3})}$

Step 7.4

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {\sqrt[4]{8 b^{3}}}^{1 + 3}}$

Step 7.5

Add $1$ and $3$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {\sqrt[4]{8 b^{3}}}^{4}}$

Step 7.6

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

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Step 7.6.1

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {({(8 b^{3})}^{\frac{1}{4}})}^{4}}$

Step 7.6.2

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {(8 b^{3})}^{\frac{1}{4} \cdot 4}}$

Step 7.6.3

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {(8 b^{3})}^{\frac{4}{4}}}$

Step 7.6.4

Cancel the common factor of $4$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {(8 b^{3})}^{\frac{4}{4}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b {(8 b^{3})}^{1}}$

Step 7.6.5

Simplify.

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b (8 b^{3})}$

Step 8

Multiply $b$ by $b^{3}$ by adding the exponents.

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Step 8.1

Move $b^{3}$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b^{3} b \cdot 8}$

Step 8.2

Multiply $b^{3}$ by $b$ .

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Step 8.2.1

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b^{3} b^{1} \cdot 8}$

Step 8.2.2

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b^{3 + 1} \cdot 8}$

Step 8.3

Add $3$ and $1$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{b^{4} \cdot 8}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{{(8 b^{3})}^{3}}}{b^{4} \cdot 8}$

Step 9.2

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{8^{3} {(b^{3})}^{3}}}{b^{4} \cdot 8}$

Step 9.3

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 {(b^{3})}^{3}}}{b^{4} \cdot 8}$

Step 9.4

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

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Step 9.4.1

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{3 \cdot 3}}}{b^{4} \cdot 8}$

Step 9.4.2

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{9}}}{b^{4} \cdot 8}$

Step 9.5

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

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Step 9.5.1

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{256 (2) b^{9}}}{b^{4} \cdot 8}$

Step 9.5.2

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} \cdot 2 b^{9}}}{b^{4} \cdot 8}$

Step 9.5.3

Factor out $b^{8}$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} \cdot 2 (b^{8} b)}}{b^{4} \cdot 8}$

Step 9.5.4

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} \cdot 2 ({(b^{2})}^{4} b)}}{b^{4} \cdot 8}$

Step 9.5.5

Move $2$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} {(b^{2})}^{4} \cdot 2 b}}{b^{4} \cdot 8}$

Step 9.5.6

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{{(4 b^{2})}^{4} \cdot 2 b}}{b^{4} \cdot 8}$

Step 9.5.7

Add parentheses.

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{{(4 b^{2})}^{4} \cdot (2 b)}}{b^{4} \cdot 8}$

Step 9.6

Pull terms out from under the radical.

$\frac{\sqrt[4]{3 a^{2}} \cdot 4 b^{2} \sqrt[4]{2 b}}{b^{4} \cdot 8}$

Step 9.7

Combine exponents.

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Step 9.7.1

Combine using the product rule for radicals.

$\frac{4 b^{2} \sqrt[4]{3 a^{2} (2 b)}}{b^{4} \cdot 8}$

Step 9.7.2

Multiply $2$ by $3$ .

$\frac{4 b^{2} \sqrt[4]{6 a^{2} b}}{b^{4} \cdot 8}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $4$ and $8$ .

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Step 10.1.1

Factor $4$ out of $4 b^{2} \sqrt[4]{6 a^{2} b}$ .

$\frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{b^{4} \cdot 8}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $4$ out of $b^{4} \cdot 8$ .

$\frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{4 (b^{4} \cdot 2)}$

Step 10.1.2.2

Cancel the common factor.

$\frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{4 (b^{4} \cdot 2)}$

Step 10.1.2.3

Rewrite the expression.

$\frac{b^{2} \sqrt[4]{6 a^{2} b}}{b^{4} \cdot 2}$

Step 10.2

Cancel the common factor of $b^{2}$ and $b^{4}$ .

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Step 10.2.1

Factor $b^{2}$ out of $b^{2} \sqrt[4]{6 a^{2} b}$ .

$\frac{b^{2} (\sqrt[4]{6 a^{2} b})}{b^{4} \cdot 2}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $b^{2}$ out of $b^{4} \cdot 2$ .

$\frac{b^{2} (\sqrt[4]{6 a^{2} b})}{b^{2} (b^{2} \cdot 2)}$

Step 10.2.2.2

Cancel the common factor.

$\frac{b^{2} \sqrt[4]{6 a^{2} b}}{b^{2} (b^{2} \cdot 2)}$

Step 10.2.2.3

Rewrite the expression.

$\frac{\sqrt[4]{6 a^{2} b}}{b^{2} \cdot 2}$

Step 10.3

Move $2$ to the left of $b^{2}$ .

$\frac{\sqrt[4]{6 a^{2} b}}{2 b^{2}}$