Simplify the Radical Expression ( square root of 2)/( cube root of 2)

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{2}}^{2}$ as ${(2^{2})}^{\frac{1}{3}}$ .

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

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

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

Step 4

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $3$ and $4$ .

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

Step 5

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.

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

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

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

Step 6

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 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{2}$

Decimal Form:

$1.12246204 \dots$

Simplify the Radical Expression ( square root of 2)/( cube root of 2)

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