Finite Math Examples

$\frac{\sqrt[3]{32 a^{7} b^{7}}}{\sqrt{2 a b}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $32 a^{7} b^{7}$ as ${(2 a^{2} b^{2})}^{3} \cdot (4 a b)$ .

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Step 1.1.1

Factor $8$ out of $32$ .

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

Step 1.1.2

Rewrite $8$ as $2^{3}$ .

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

Step 1.1.3

Factor out $a^{6}$ .

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

Step 1.1.4

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

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

Step 1.1.5

Factor out $b^{6}$ .

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

Step 1.1.6

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

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

Step 1.1.7

Move $a$ .

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

Step 1.1.8

Move $4$ .

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

Step 1.1.9

Rewrite $(2^{3} ({(a^{2})}^{3})) {(b^{2})}^{3}$ as ${(2 a^{2} b^{2})}^{3}$ .

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

Step 1.1.10

Add parentheses.

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

Step 1.1.11

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $a$ out of $a b$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.2.5

Divide $a b \sqrt[3]{4 a b} \sqrt{2 a b}$ by $1$ .

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

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

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

Step 7.1.5

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

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

Step 7.1.6

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

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

Step 7.2

Combine using the product rule for radicals.

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

Step 7.3

Apply the product rule to $2 a b$ .

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

Step 7.4

Apply the product rule to $4 a b$ .

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

Step 7.5

Multiply $b^{3}$ by $b^{2}$ by adding the exponents.

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Step 7.5.1

Move $b^{2}$ .

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

Step 7.5.2

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

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

Step 7.5.3

Add $2$ and $3$ .

$a b \sqrt[6]{{(2 a)}^{3} b^{5} {(4 a)}^{2}}$

Step 7.6

Apply the product rule to $2 a$ .

$a b \sqrt[6]{2^{3} a^{3} b^{5} {(4 a)}^{2}}$

Step 7.7

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

$a b \sqrt[6]{8 a^{3} b^{5} {(4 a)}^{2}}$

Step 7.8

Apply the product rule to $4 a$ .

$a b \sqrt[6]{8 a^{3} b^{5} (4^{2} a^{2})}$

Step 7.9

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

$a b \sqrt[6]{8 a^{3} b^{5} \cdot 16 a^{2}}$

Step 7.10

Combine exponents.

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Step 7.10.1

Multiply $16$ by $8$ .

$a b \sqrt[6]{128 a^{3} b^{5} a^{2}}$

Step 7.10.2

Multiply $a^{3}$ by $a^{2}$ by adding the exponents.

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Step 7.10.2.1

Move $a^{2}$ .

$a b \sqrt[6]{128 (a^{2} a^{3}) b^{5}}$

Step 7.10.2.2

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

$a b \sqrt[6]{128 a^{2 + 3} b^{5}}$

Step 7.10.2.3

Add $2$ and $3$ .

$a b \sqrt[6]{128 a^{5} b^{5}}$

Step 8

Rewrite $128 a^{5} b^{5}$ as $2^{6} \cdot (2 a^{5} b^{5})$ .

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Step 8.1

Factor $64$ out of $128$ .

$a b \sqrt[6]{64 (2) a^{5} b^{5}}$

Step 8.2

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

$a b \sqrt[6]{2^{6} \cdot 2 a^{5} b^{5}}$

Step 8.3

Add parentheses.

$a b \sqrt[6]{2^{6} \cdot 2 (a^{5} b^{5})}$

Step 8.4

Add parentheses.

$a b \sqrt[6]{2^{6} \cdot (2 a^{5} b^{5})}$

Step 9

Pull terms out from under the radical.

$a b (2 \sqrt[6]{2 a^{5} b^{5}})$

Step 10

Rewrite using the commutative property of multiplication.

$2 a b \sqrt[6]{2 a^{5} b^{5}}$