Simplify cube root of 40/125

$\sqrt[3]{\frac{40}{125}}$

Step 1

Cancel the common factor of $40$ and $125$ .

Factor $5$ out of $40$ .

$\sqrt[3]{\frac{5 (8)}{125}}$

Cancel the common factors.

Factor $5$ out of $125$ .

$\sqrt[3]{\frac{5 \cdot 8}{5 \cdot 25}}$

Cancel the common factor.

$\sqrt[3]{\frac{5 \cdot 8}{5 \cdot 25}}$

Rewrite the expression.

$\sqrt[3]{\frac{8}{25}}$

Step 2

Rewrite $\sqrt[3]{\frac{8}{25}}$ as $\frac{\sqrt[3]{8}}{\sqrt[3]{25}}$ .

$\frac{\sqrt[3]{8}}{\sqrt[3]{25}}$

Step 3

Simplify the numerator.

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

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

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

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

Step 4

Multiply $\frac{2}{\sqrt[3]{25}}$ by $\frac{{\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{2}}$ .

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

Step 5

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt[3]{25}}$ by $\frac{{\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{2}}$ .

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

Raise $\sqrt[3]{25}$ to the power of $1$ .

$\frac{2 {\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{1} {\sqrt[3]{25}}^{2}}$

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

$\frac{2 {\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{1 + 2}}$

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{25}}^{3}$ as $25$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$\frac{2 {\sqrt[3]{25}}^{2}}{{(25^{\frac{1}{3}})}^{3}}$

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

$\frac{2 {\sqrt[3]{25}}^{2}}{25^{\frac{1}{3} \cdot 3}}$

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 {\sqrt[3]{25}}^{2}}{25^{1}}$

Evaluate the exponent.

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

Step 6

Simplify the numerator.

Rewrite ${\sqrt[3]{25}}^{2}$ as ${(25^{2})}^{\frac{1}{3}}$ .

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

Raise $25$ to the power of $2$ .

$\frac{2 \sqrt[3]{625}}{25}$

Rewrite $625$ as $5^{3} \cdot 5$ .

Factor $125$ out of $625$ .

$\frac{2 \sqrt[3]{125 (5)}}{25}$

Rewrite $125$ as $5^{3}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $5$ .

$\frac{10 \sqrt[3]{5}}{25}$

Step 7

Cancel the common factor of $10$ and $25$ .

Factor $5$ out of $10 \sqrt[3]{5}$ .

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

Cancel the common factors.

Factor $5$ out of $25$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.68399037 \dots$

Simplify cube root of 40/125

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