Pre-Algebra Examples

${(\sqrt{\frac{49}{8}})}^{3}$

Step 1

Rewrite $\sqrt{\frac{49}{8}}$ as $\frac{\sqrt{49}}{\sqrt{8}}$ .

${(\frac{\sqrt{49}}{\sqrt{8}})}^{3}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

${(\frac{\sqrt{7^{2}}}{\sqrt{8}})}^{3}$

Step 2.2

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

${(\frac{7}{\sqrt{8}})}^{3}$

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

Factor $4$ out of $8$ .

${(\frac{7}{\sqrt{4 (2)}})}^{3}$

Step 3.1.2

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

${(\frac{7}{\sqrt{2^{2} \cdot 2}})}^{3}$

Step 3.2

Pull terms out from under the radical.

${(\frac{7}{2 \sqrt{2}})}^{3}$

Step 4

Multiply $\frac{7}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{7}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{3}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{7}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{7 \sqrt{2}}{2 \sqrt{2} \sqrt{2}})}^{3}$

Step 5.2

Move $\sqrt{2}$ .

${(\frac{7 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}^{3}$

Step 5.3

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

${(\frac{7 \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})})}^{3}$

Step 5.4

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

${(\frac{7 \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})}^{3}$

Step 5.5

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

${(\frac{7 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}})}^{3}$

Step 5.6

Add $1$ and $1$ .

${(\frac{7 \sqrt{2}}{2 {\sqrt{2}}^{2}})}^{3}$

Step 5.7

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

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Step 5.7.1

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

${(\frac{7 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}})}^{3}$

Step 5.7.2

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

${(\frac{7 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}})}^{3}$

Step 5.7.3

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

${(\frac{7 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}^{3}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

${(\frac{7 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}^{3}$

Step 5.7.4.2

Rewrite the expression.

${(\frac{7 \sqrt{2}}{2 \cdot 2^{1}})}^{3}$

Step 5.7.5

Evaluate the exponent.

${(\frac{7 \sqrt{2}}{2 \cdot 2})}^{3}$

Step 6

Multiply $2$ by $2$ .

${(\frac{7 \sqrt{2}}{4})}^{3}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to $\frac{7 \sqrt{2}}{4}$ .

$\frac{{(7 \sqrt{2})}^{3}}{4^{3}}$

Step 7.2

Apply the product rule to $7 \sqrt{2}$ .

$\frac{7^{3} {\sqrt{2}}^{3}}{4^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $7$ to the power of $3$ .

$\frac{343 {\sqrt{2}}^{3}}{4^{3}}$

Step 8.2

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\frac{343 \sqrt{2^{3}}}{4^{3}}$

Step 8.3

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

$\frac{343 \sqrt{8}}{4^{3}}$

Step 8.4

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

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Step 8.4.1

Factor $4$ out of $8$ .

$\frac{343 \sqrt{4 (2)}}{4^{3}}$

Step 8.4.2

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

$\frac{343 \sqrt{2^{2} \cdot 2}}{4^{3}}$

Step 8.5

Pull terms out from under the radical.

$\frac{343 \cdot 2 \sqrt{2}}{4^{3}}$

Step 8.6

Multiply $343$ by $2$ .

$\frac{686 \sqrt{2}}{4^{3}}$

Step 9

Raise $4$ to the power of $3$ .

$\frac{686 \sqrt{2}}{64}$

Step 10

Cancel the common factor of $686$ and $64$ .

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Step 10.1

Factor $2$ out of $686 \sqrt{2}$ .

$\frac{2 (343 \sqrt{2})}{64}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $64$ .

$\frac{2 (343 \sqrt{2})}{2 (32)}$

Step 10.2.2

Cancel the common factor.

$\frac{2 (343 \sqrt{2})}{2 \cdot 32}$

Step 10.2.3

Rewrite the expression.

$\frac{343 \sqrt{2}}{32}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{343 \sqrt{2}}{32}$

Decimal Form:

$15.15860162 \dots$