Simplify square root of cube root of 2/5

$\sqrt{\sqrt[3]{\frac{2}{5}}}$

Step 1

Rewrite $\sqrt[3]{\frac{2}{5}}$ as $\frac{\sqrt[3]{2}}{\sqrt[3]{5}}$ .

$\sqrt{\frac{\sqrt[3]{2}}{\sqrt[3]{5}}}$

Step 2

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

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

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

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

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\sqrt{\frac{\sqrt[3]{2} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}}$

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

$\sqrt{\frac{\sqrt[3]{2} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}}$

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\sqrt{\frac{\sqrt[3]{2} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}}$

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

$\sqrt{\frac{\sqrt[3]{2} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}}$

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{\sqrt[3]{2} {\sqrt[3]{5}}^{2}}{5^{1}}}$

Evaluate the exponent.

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

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{5}}^{2}$ as ${(5^{2})}^{\frac{1}{3}}$ .

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

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

$\sqrt{\frac{\sqrt[3]{2} \sqrt[3]{25}}{5}}$

Step 5

Simplify the numerator.

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt[3]{2 \cdot 25}}{5}}$

Multiply $2$ by $25$ .

$\sqrt{\frac{\sqrt[3]{50}}{5}}$

Step 6

Rewrite $\sqrt{\frac{\sqrt[3]{50}}{5}}$ as $\frac{\sqrt{\sqrt[3]{50}}}{\sqrt{5}}$ .

$\frac{\sqrt{\sqrt[3]{50}}}{\sqrt{5}}$

Step 7

Rewrite $\sqrt{\sqrt[3]{50}}$ as $\sqrt[6]{50}$ .

$\frac{\sqrt[6]{50}}{\sqrt{5}}$

Step 8

Multiply $\frac{\sqrt[6]{50}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[6]{50}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9

Combine and simplify the denominator.

Multiply $\frac{\sqrt[6]{50}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[6]{50} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[6]{50} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[6]{50} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{\sqrt[6]{50} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt[6]{50} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[6]{50} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt[6]{50} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt[6]{50} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt[6]{50} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt[6]{50} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$\frac{\sqrt[6]{50} \sqrt{5}}{5}$

Step 10

Simplify the numerator.

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[6]{50} \cdot 5^{\frac{1}{2}}}{5}$

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

$\frac{\sqrt[6]{50} \cdot 5^{\frac{3}{6}}}{5}$

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

$\frac{\sqrt[6]{50} \sqrt[6]{5^{3}}}{5}$

Combine using the product rule for radicals.

$\frac{\sqrt[6]{50 \cdot 5^{3}}}{5}$

Raise $5$ to the power of $3$ .

$\frac{\sqrt[6]{50 \cdot 125}}{5}$

Step 11

Multiply $50$ by $125$ .

$\frac{\sqrt[6]{6250}}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{6250}}{5}$

Decimal Form:

$0.85837421 \dots$

Simplify square root of cube root of 2/5

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