Basic Math Examples

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

Step 1

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

${\sqrt[6]{2 \sqrt{2}}}^{8}$

Step 2

Rewrite ${\sqrt[6]{2 \sqrt{2}}}^{8}$ as $\sqrt[6]{{(2 \sqrt{2})}^{8}}$ .

$\sqrt[6]{{(2 \sqrt{2})}^{8}}$

Step 3

Simplify the expression.

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Step 3.1

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

$\sqrt[6]{2^{8} {\sqrt{2}}^{8}}$

Step 3.2

Raise $2$ to the power of $8$ .

$\sqrt[6]{256 {\sqrt{2}}^{8}}$

Step 4

Rewrite ${\sqrt{2}}^{8}$ as $2^{4}$ .

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Step 4.1

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

$\sqrt[6]{256 {(2^{\frac{1}{2}})}^{8}}$

Step 4.2

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

$\sqrt[6]{256 \cdot 2^{\frac{1}{2} \cdot 8}}$

Step 4.3

Combine $\frac{1}{2}$ and $8$ .

$\sqrt[6]{256 \cdot 2^{\frac{8}{2}}}$

Step 4.4

Cancel the common factor of $8$ and $2$ .

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Step 4.4.1

Factor $2$ out of $8$ .

$\sqrt[6]{256 \cdot 2^{\frac{2 \cdot 4}{2}}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $2$ .

$\sqrt[6]{256 \cdot 2^{\frac{2 \cdot 4}{2 (1)}}}$

Step 4.4.2.2

Cancel the common factor.

$\sqrt[6]{256 \cdot 2^{\frac{2 \cdot 4}{2 \cdot 1}}}$

Step 4.4.2.3

Rewrite the expression.

$\sqrt[6]{256 \cdot 2^{\frac{4}{1}}}$

Step 4.4.2.4

Divide $4$ by $1$ .

$\sqrt[6]{256 \cdot 2^{4}}$

Step 5

Simplify the expression.

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Step 5.1

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

$\sqrt[6]{256 \cdot 16}$

Step 5.2

Multiply $256$ by $16$ .

$\sqrt[6]{4096}$

Step 5.3

Rewrite $4096$ as $4^{6}$ .

$\sqrt[6]{4^{6}}$

Step 5.4

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

$4$