Precalculus Examples

$\frac{\sqrt{8} \sqrt[3]{3}}{\sqrt[3]{24 \sqrt{8}}}$

Step 1

Combine $\sqrt[3]{3}$ and $\sqrt[3]{24 \sqrt{8}}$ into a single radical.

$\sqrt{8} \sqrt[3]{\frac{3}{24 \sqrt{8}}}$

Step 2

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} \sqrt[3]{\frac{3}{24 \sqrt{8}}}$

Step 2.2

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

$\sqrt{2^{2} \cdot 2} \sqrt[3]{\frac{3}{24 \sqrt{8}}}$

Step 3

Simplify terms.

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Step 3.1

Pull terms out from under the radical.

$2 \sqrt{2} \sqrt[3]{\frac{3}{24 \sqrt{8}}}$

Step 3.2

Reduce the expression $\frac{3}{24 \sqrt{8}}$ by cancelling the common factors.

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Step 3.2.1

Factor $3$ out of $3$ .

$2 \sqrt{2} \sqrt[3]{\frac{3 (1)}{24 \sqrt{8}}}$

Step 3.2.2

Factor $3$ out of $24 \sqrt{8}$ .

$2 \sqrt{2} \sqrt[3]{\frac{3 (1)}{3 (8 \sqrt{8})}}$

Step 3.2.3

Cancel the common factor.

$2 \sqrt{2} \sqrt[3]{\frac{3 \cdot 1}{3 (8 \sqrt{8})}}$

Step 3.2.4

Rewrite the expression.

$2 \sqrt{2} \sqrt[3]{\frac{1}{8 \sqrt{8}}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.1

Factor $4$ out of $8$ .

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

Step 4.1.2

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

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

Step 4.2

Pull terms out from under the radical.

$2 \sqrt{2} \sqrt[3]{\frac{1}{8 \cdot 2 \sqrt{2}}}$

Step 4.3

Multiply $8$ by $2$ .

$2 \sqrt{2} \sqrt[3]{\frac{1}{16 \sqrt{2}}}$

Step 5

Multiply $\frac{1}{16 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$2 \sqrt{2} \sqrt[3]{\frac{1}{16 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{1}{16 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \sqrt{2} \sqrt{2}}}$

Step 6.2

Move $\sqrt{2}$ .

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 (\sqrt{2} \sqrt{2})}}$

Step 6.3

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 ({\sqrt{2}}^{1} \sqrt{2})}}$

Step 6.4

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}}$

Step 6.5

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 {\sqrt{2}}^{1 + 1}}}$

Step 6.6

Add $1$ and $1$ .

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 {\sqrt{2}}^{2}}}$

Step 6.7

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

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Step 6.7.1

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 {(2^{\frac{1}{2}})}^{2}}}$

Step 6.7.2

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \cdot 2^{\frac{1}{2} \cdot 2}}}$

Step 6.7.3

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

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \cdot 2^{\frac{2}{2}}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \cdot 2^{\frac{2}{2}}}}$

Step 6.7.4.2

Rewrite the expression.

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \cdot 2^{1}}}$

Step 6.7.5

Evaluate the exponent.

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{16 \cdot 2}}$

Step 7

Multiply $16$ by $2$ .

$2 \sqrt{2} \sqrt[3]{\frac{\sqrt{2}}{32}}$

Step 8

Rewrite $\sqrt[3]{\frac{\sqrt{2}}{32}}$ as $\frac{\sqrt[3]{\sqrt{2}}}{\sqrt[3]{32}}$ .

$2 \sqrt{2} \frac{\sqrt[3]{\sqrt{2}}}{\sqrt[3]{32}}$

Step 9

Rewrite $\sqrt[3]{\sqrt{2}}$ as $\sqrt[6]{2}$ .

$2 \sqrt{2} \frac{\sqrt[6]{2}}{\sqrt[3]{32}}$

Step 10

Simplify the denominator.

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Step 10.1

Rewrite $32$ as $2^{3} \cdot 4$ .

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Step 10.1.1

Factor $8$ out of $32$ .

$2 \sqrt{2} \frac{\sqrt[6]{2}}{\sqrt[3]{8 (4)}}$

Step 10.1.2

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

$2 \sqrt{2} \frac{\sqrt[6]{2}}{\sqrt[3]{2^{3} \cdot 4}}$

Step 10.2

Pull terms out from under the radical.

$2 \sqrt{2} \frac{\sqrt[6]{2}}{2 \sqrt[3]{4}}$

Step 11

Simplify terms.

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Step 11.1

Cancel the common factor of $2$ .

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Step 11.1.1

Factor $2$ out of $2 \sqrt{2}$ .

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

Step 11.1.2

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

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

Step 11.1.3

Cancel the common factor.

$2 \sqrt{2} \frac{\sqrt[6]{2}}{2 \sqrt[3]{4}}$

Step 11.1.4

Rewrite the expression.

$\sqrt{2} \frac{\sqrt[6]{2}}{\sqrt[3]{4}}$

Step 11.2

Combine $\sqrt{2}$ and $\frac{\sqrt[6]{2}}{\sqrt[3]{4}}$ .

$\frac{\sqrt{2} \sqrt[6]{2}}{\sqrt[3]{4}}$

Step 12

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

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Step 12.1

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

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

Step 12.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

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

Step 12.3

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

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

Step 13

Combine using the product rule for radicals.

$\frac{\sqrt[6]{2^{3} \cdot 2}}{\sqrt[3]{4}}$

Step 14

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

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Step 14.1

Multiply $2^{3}$ by $2$ .

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Step 14.1.1

Raise $2$ to the power of $1$ .

$\frac{\sqrt[6]{2^{3} \cdot 2^{1}}}{\sqrt[3]{4}}$

Step 14.1.2

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

$\frac{\sqrt[6]{2^{3 + 1}}}{\sqrt[3]{4}}$

Step 14.2

Add $3$ and $1$ .

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

Step 15

Simplify the numerator.

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Step 15.1

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

$\frac{\sqrt[6]{16}}{\sqrt[3]{4}}$

Step 15.2

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

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

Step 15.3

Rewrite $\sqrt[6]{4^{2}}$ as $\sqrt[3]{\sqrt{4^{2}}}$ .

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

Step 15.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt[3]{4}}{\sqrt[3]{4}}$

Step 16

Cancel the common factor of $\sqrt[3]{4}$ .

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Step 16.1

Cancel the common factor.

$\frac{\sqrt[3]{4}}{\sqrt[3]{4}}$

Step 16.2

Rewrite the expression.

$1$