Simplify the Radical Expression cube root of 2^5* square root of 2^3

$\sqrt[3]{2^{5}} \cdot \sqrt{2^{3}}$

Step 1

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

$\sqrt[3]{32} \cdot \sqrt{2^{3}}$

Step 2

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

Factor $8$ out of $32$ .

$\sqrt[3]{8 (4)} \cdot \sqrt{2^{3}}$

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

$2 \sqrt[3]{4} \cdot \sqrt{8}$

Step 5

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

Factor $4$ out of $8$ .

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

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

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

Step 6

Pull terms out from under the radical.

$2 \sqrt[3]{4} \cdot (| 2 | \sqrt{2})$

Step 7

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 8

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

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

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

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

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

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

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

Combine using the product rule for radicals.

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

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

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

Multiply the exponents in ${(2^{2})}^{2}$ .

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

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

Multiply $2$ by $2$ .

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

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

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

Add $3$ and $4$ .

$4 \sqrt[6]{2^{7}}$

Step 9

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

$4 \sqrt[6]{128}$

Step 10

Rewrite $128$ as $2^{6} \cdot 2$ .

Factor $64$ out of $128$ .

$4 \sqrt[6]{64 (2)}$

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 13

Multiply $2$ by $4$ .

$8 \sqrt[6]{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt[6]{2}$

Decimal Form:

$8.97969638 \dots$

Simplify the Radical Expression cube root of 2^5* square root of 2^3

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