Evaluate ( cube root of 4)/( square root of 2)

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Simplify the numerator.

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

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

$\frac{4^{\frac{1}{3}} \sqrt{2}}{2}$

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

$\frac{4^{\frac{2}{6}} \sqrt{2}}{2}$

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $4$ and $3$ .

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

Step 4

Simplify the numerator.

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

$\frac{\sqrt[6]{128}}{2}$

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

Factor $64$ out of $128$ .

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

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

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

Pull terms out from under the radical.

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

Step 5

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\sqrt[6]{2}$ by $1$ .

$\sqrt[6]{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{2}$

Decimal Form:

$1.12246204 \dots$

Evaluate ( cube root of 4)/( square root of 2)

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