Simplify fourth root of 8* square root of 2* sixth root of 4

$\sqrt[4]{8} \cdot \sqrt{2} \cdot \sqrt[6]{4}$

Step 1

Multiply $\sqrt[4]{8} \sqrt{2}$ .

Rewrite the expression using the least common index of $4$ .

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

$\sqrt[4]{8} \cdot 2^{\frac{1}{2}} \cdot \sqrt[6]{4}$

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$\sqrt[4]{8} \cdot 2^{\frac{2}{4}} \cdot \sqrt[6]{4}$

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

$\sqrt[4]{8} \sqrt[4]{2^{2}} \cdot \sqrt[6]{4}$

Combine using the product rule for radicals.

$\sqrt[4]{8 \cdot 2^{2}} \cdot \sqrt[6]{4}$

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

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

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

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

Add $3$ and $2$ .

$\sqrt[4]{2^{5}} \cdot \sqrt[6]{4}$

Step 2

Raise $2$ to the power of $5$ .

$\sqrt[4]{32} \cdot \sqrt[6]{4}$

Step 3

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

Factor $16$ out of $32$ .

$\sqrt[4]{16 (2)} \cdot \sqrt[6]{4}$

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

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

$2 \sqrt[4]{2} \cdot \sqrt[3]{2}$

Step 8

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

Rewrite the expression using the least common index of $12$ .

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

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

Rewrite $2^{\frac{1}{3}}$ as $2^{\frac{4}{12}}$ .

$2 (2^{\frac{4}{12}} \sqrt[4]{2})$

Rewrite $2^{\frac{4}{12}}$ as $\sqrt[12]{2^{4}}$ .

$2 (\sqrt[12]{2^{4}} \sqrt[4]{2})$

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

$2 (\sqrt[12]{2^{4}} \cdot 2^{\frac{1}{4}})$

Rewrite $2^{\frac{1}{4}}$ as $2^{\frac{3}{12}}$ .

$2 (\sqrt[12]{2^{4}} \cdot 2^{\frac{3}{12}})$

Rewrite $2^{\frac{3}{12}}$ as $\sqrt[12]{2^{3}}$ .

$2 (\sqrt[12]{2^{4}} \sqrt[12]{2^{3}})$

Combine using the product rule for radicals.

$2 \sqrt[12]{2^{4} \cdot 2^{3}}$

Multiply $2^{4}$ by $2^{3}$ by adding the exponents.

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

$2 \sqrt[12]{2^{4 + 3}}$

Add $4$ and $3$ .

$2 \sqrt[12]{2^{7}}$

Step 9

Raise $2$ to the power of $7$ .

$2 \sqrt[12]{128}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[12]{128}$

Decimal Form:

$2.99661415 \dots$

Simplify fourth root of 8* square root of 2* sixth root of 4

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