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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $3$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $4$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

$| \frac{1}{b} | \frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 {(b^{3})}^{3}}}{8 b^{3}}$

Step 6.4

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

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Step 6.4.1

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

$| \frac{1}{b} | \frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{3 \cdot 3}}}{8 b^{3}}$

Step 6.4.2

Multiply $3$ by $3$ .

$| \frac{1}{b} | \frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{9}}}{8 b^{3}}$

Step 6.5

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

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Step 6.5.1

Factor $256$ out of $512$ .

$| \frac{1}{b} | \frac{\sqrt[4]{3 a^{2}} \sqrt[4]{256 (2) b^{9}}}{8 b^{3}}$

Step 6.5.2

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

$| \frac{1}{b} | \frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} \cdot 2 b^{9}}}{8 b^{3}}$

Step 6.5.3

Factor out $b^{8}$ .

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

Step 6.5.4

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

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

Step 6.5.5

Move $2$ .

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

Step 6.5.6

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

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

Step 6.5.7

Add parentheses.

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

Step 6.6

Pull terms out from under the radical.

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

Step 6.7

Remove non-negative terms from the absolute value.

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

Step 6.8

Combine exponents.

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Step 6.8.1

Combine using the product rule for radicals.

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

Step 6.8.2

Multiply $2$ by $3$ .

$| \frac{1}{b} | \frac{4 b^{2} \sqrt[4]{6 a^{2} b}}{8 b^{3}}$

Step 7

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

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Step 7.1

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

$| \frac{1}{b} | \frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{8 b^{3}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $4$ out of $8 b^{3}$ .

$| \frac{1}{b} | \frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{4 (2 b^{3})}$

Step 7.2.2

Cancel the common factor.

$| \frac{1}{b} | \frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{4 (2 b^{3})}$

Step 7.2.3

Rewrite the expression.

$| \frac{1}{b} | \frac{b^{2} \sqrt[4]{6 a^{2} b}}{2 b^{3}}$

Step 8

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

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Step 8.1

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

$| \frac{1}{b} | \frac{b^{2} (\sqrt[4]{6 a^{2} b})}{2 b^{3}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $b^{2}$ out of $2 b^{3}$ .

$| \frac{1}{b} | \frac{b^{2} (\sqrt[4]{6 a^{2} b})}{b^{2} (2 b)}$

Step 8.2.2

Cancel the common factor.

$| \frac{1}{b} | \frac{b^{2} \sqrt[4]{6 a^{2} b}}{b^{2} (2 b)}$

Step 8.2.3

Rewrite the expression.

$| \frac{1}{b} | \frac{\sqrt[4]{6 a^{2} b}}{2 b}$

Step 9

Combine $| \frac{1}{b} |$ and $\frac{\sqrt[4]{6 a^{2} b}}{2 b}$ .

$\frac{| \frac{1}{b} | \sqrt[4]{6 a^{2} b}}{2 b}$