Finite Math Examples

$\frac{\sqrt{32 a^{5} b}}{\sqrt[3]{2 a b}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $32 a^{5} b$ as ${(4 a^{2})}^{2} \cdot (2 a b)$ .

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Step 1.1.1

Factor $16$ out of $32$ .

$\frac{\sqrt{16 (2) a^{5} b}}{\sqrt[3]{2 a b}}$

Step 1.1.2

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

$\frac{\sqrt{4^{2} \cdot 2 a^{5} b}}{\sqrt[3]{2 a b}}$

Step 1.1.3

Factor out $a^{4}$ .

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

Step 1.1.4

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

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

Step 1.1.5

Move $2$ .

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

Step 1.1.6

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

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

Step 1.1.7

Add parentheses.

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

Step 1.1.8

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Add $1$ and $2$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

Cancel the common factor of $3$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Simplify.

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

Step 3.2

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

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Step 3.2.1

Factor $2$ out of $4 a^{2} \sqrt{2 a b} {\sqrt[3]{2 a b}}^{2}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $2 a b$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

Factor $a$ out of $2 a^{2} \sqrt{2 a b} {\sqrt[3]{2 a b}}^{2}$ .

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $a$ out of $a b$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{2 a b}}^{2}$ as $\sqrt[3]{{(2 a b)}^{2}}$ .

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

Step 4.2

Apply the product rule to $2 a b$ .

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

Step 4.3

Apply the product rule to $2 a$ .

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

Step 4.4

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

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

Step 4.5

Combine exponents.

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Step 4.5.1

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

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Step 4.5.1.1

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

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

Step 4.5.1.2

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

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

Step 4.5.1.3

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

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

Step 4.5.1.4

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

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

Step 4.5.1.5

Rewrite ${(2 a b)}^{\frac{1}{2}}$ as ${(2 a b)}^{\frac{3}{6}}$ .

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

Step 4.5.1.6

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

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

Step 4.5.2

Combine using the product rule for radicals.

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

Step 4.5.3

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

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

Step 4.5.4

Apply the product rule to $2 a b$ .

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

Step 4.5.5

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

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

Step 4.5.6

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

$\frac{2 a \sqrt[6]{16 {(a^{2})}^{2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 4.5.7

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

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Step 4.5.7.1

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

$\frac{2 a \sqrt[6]{16 a^{2 \cdot 2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 4.5.7.2

Multiply $2$ by $2$ .

$\frac{2 a \sqrt[6]{16 a^{4} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 4.5.8

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

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Step 4.5.8.1

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

$\frac{2 a \sqrt[6]{16 a^{4} b^{2 \cdot 2} {(2 a)}^{3} b^{3}}}{b}$

Step 4.5.8.2

Multiply $2$ by $2$ .

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} {(2 a)}^{3} b^{3}}}{b}$

Step 4.5.9

Apply the product rule to $2 a$ .

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} (2^{3} a^{3}) b^{3}}}{b}$

Step 4.5.10

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

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} \cdot 8 a^{3} b^{3}}}{b}$

Step 4.5.11

Combine exponents.

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Step 4.5.11.1

Multiply $8$ by $16$ .

$\frac{2 a \sqrt[6]{128 a^{4} b^{4} a^{3} b^{3}}}{b}$

Step 4.5.11.2

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

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Step 4.5.11.2.1

Move $a^{3}$ .

$\frac{2 a \sqrt[6]{128 (a^{3} a^{4}) b^{4} b^{3}}}{b}$

Step 4.5.11.2.2

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

$\frac{2 a \sqrt[6]{128 a^{3 + 4} b^{4} b^{3}}}{b}$

Step 4.5.11.2.3

Add $3$ and $4$ .

$\frac{2 a \sqrt[6]{128 a^{7} b^{4} b^{3}}}{b}$

Step 4.5.11.3

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

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Step 4.5.11.3.1

Move $b^{3}$ .

$\frac{2 a \sqrt[6]{128 a^{7} (b^{3} b^{4})}}{b}$

Step 4.5.11.3.2

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

$\frac{2 a \sqrt[6]{128 a^{7} b^{3 + 4}}}{b}$

Step 4.5.11.3.3

Add $3$ and $4$ .

$\frac{2 a \sqrt[6]{128 a^{7} b^{7}}}{b}$

Step 4.6

Rewrite $128 a^{7} b^{7}$ as ${(2 a b)}^{6} \cdot (2 a b)$ .

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Step 4.6.1

Factor $64$ out of $128$ .

$\frac{2 a \sqrt[6]{64 (2) a^{7} b^{7}}}{b}$

Step 4.6.2

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

$\frac{2 a \sqrt[6]{2^{6} \cdot 2 a^{7} b^{7}}}{b}$

Step 4.6.3

Factor out $a^{6}$ .

$\frac{2 a \sqrt[6]{2^{6} \cdot 2 (a^{6} a) b^{7}}}{b}$

Step 4.6.4

Factor out $b^{6}$ .

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

Step 4.6.5

Move $a$ .

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

Step 4.6.6

Move $2$ .

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

Step 4.6.7

Rewrite $(2^{6} (a^{6})) b^{6}$ as ${(2 a b)}^{6}$ .

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

Step 4.6.8

Add parentheses.

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

Step 4.6.9

Add parentheses.

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

Step 4.7

Pull terms out from under the radical.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $2$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite.

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

Step 6.2

Factor $2$ out of $4$ .

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

Step 6.3

Rewrite.

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

Step 6.4

Divide $4$ by $2$ .

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

Step 6.5

Factor $a$ out of $a^{2}$ .

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

Step 6.6

Rewrite.

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

Step 6.7

Simplify.

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

Step 6.8

Rewrite ${\sqrt[3]{2 a b}}^{2}$ as $\sqrt[3]{{(2 a b)}^{2}}$ .

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

Step 6.9

Apply the product rule to $2 a b$ .

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

Step 6.10

Apply the product rule to $2 a$ .

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

Step 6.11

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

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

Step 6.12

Combine exponents.

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Step 6.12.1

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

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Step 6.12.1.1

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

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

Step 6.12.1.2

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

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

Step 6.12.1.3

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

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

Step 6.12.1.4

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

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

Step 6.12.1.5

Rewrite ${(2 a b)}^{\frac{1}{2}}$ as ${(2 a b)}^{\frac{3}{6}}$ .

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

Step 6.12.1.6

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

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

Step 6.12.2

Combine using the product rule for radicals.

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

Step 6.12.3

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

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

Step 6.12.4

Apply the product rule to $2 a b$ .

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

Step 6.12.5

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

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

Step 6.12.6

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

$\frac{2 a \sqrt[6]{16 {(a^{2})}^{2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 6.12.7

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

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Step 6.12.7.1

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

$\frac{2 a \sqrt[6]{16 a^{2 \cdot 2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 6.12.7.2

Multiply $2$ by $2$ .

$\frac{2 a \sqrt[6]{16 a^{4} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}}{b}$

Step 6.12.8

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

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Step 6.12.8.1

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

$\frac{2 a \sqrt[6]{16 a^{4} b^{2 \cdot 2} {(2 a)}^{3} b^{3}}}{b}$

Step 6.12.8.2

Multiply $2$ by $2$ .

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} {(2 a)}^{3} b^{3}}}{b}$

Step 6.12.9

Apply the product rule to $2 a$ .

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} (2^{3} a^{3}) b^{3}}}{b}$

Step 6.12.10

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

$\frac{2 a \sqrt[6]{16 a^{4} b^{4} \cdot 8 a^{3} b^{3}}}{b}$

Step 6.12.11

Combine exponents.

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Step 6.12.11.1

Multiply $8$ by $16$ .

$\frac{2 a \sqrt[6]{128 a^{4} b^{4} a^{3} b^{3}}}{b}$

Step 6.12.11.2

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

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Step 6.12.11.2.1

Move $a^{3}$ .

$\frac{2 a \sqrt[6]{128 (a^{3} a^{4}) b^{4} b^{3}}}{b}$

Step 6.12.11.2.2

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

$\frac{2 a \sqrt[6]{128 a^{3 + 4} b^{4} b^{3}}}{b}$

Step 6.12.11.2.3

Add $3$ and $4$ .

$\frac{2 a \sqrt[6]{128 a^{7} b^{4} b^{3}}}{b}$

Step 6.12.11.3

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

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Step 6.12.11.3.1

Move $b^{3}$ .

$\frac{2 a \sqrt[6]{128 a^{7} (b^{3} b^{4})}}{b}$

Step 6.12.11.3.2

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

$\frac{2 a \sqrt[6]{128 a^{7} b^{3 + 4}}}{b}$

Step 6.12.11.3.3

Add $3$ and $4$ .

$\frac{2 a \sqrt[6]{128 a^{7} b^{7}}}{b}$

Step 6.13

Rewrite $128 a^{7} b^{7}$ as ${(2 a b)}^{6} \cdot (2 a b)$ .

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Step 6.13.1

Factor $64$ out of $128$ .

$\frac{2 a \sqrt[6]{64 (2) a^{7} b^{7}}}{b}$

Step 6.13.2

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

$\frac{2 a \sqrt[6]{2^{6} \cdot 2 a^{7} b^{7}}}{b}$

Step 6.13.3

Factor out $a^{6}$ .

$\frac{2 a \sqrt[6]{2^{6} \cdot 2 (a^{6} a) b^{7}}}{b}$

Step 6.13.4

Factor out $b^{6}$ .

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

Step 6.13.5

Move $a$ .

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

Step 6.13.6

Move $2$ .

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

Step 6.13.7

Rewrite $(2^{6} (a^{6})) b^{6}$ as ${(2 a b)}^{6}$ .

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

Step 6.13.8

Add parentheses.

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

Step 6.13.9

Add parentheses.

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

Step 6.14

Pull terms out from under the radical.

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

Step 6.15

Multiply $2$ by $2$ .

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

Step 6.16

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

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

Step 6.17

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

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

Step 6.18

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

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

Step 6.19

Add $1$ and $1$ .

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

Step 6.20

Remove unnecessary parentheses.

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

Step 7

Cancel the common factor of $b$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $4 a^{2} \sqrt[6]{2 a b}$ by $1$ .

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