Simplify the Radical Expression ( cube root of 24)/( cube root of 15)

$\frac{\sqrt[3]{24}}{\sqrt[3]{15}}$

Step 1

Combine $\sqrt[3]{24}$ and $\sqrt[3]{15}$ into a single radical.

$\sqrt[3]{\frac{24}{15}}$

Step 2

Cancel the common factor of $24$ and $15$ .

Factor $3$ out of $24$ .

$\sqrt[3]{\frac{3 (8)}{15}}$

Cancel the common factors.

Factor $3$ out of $15$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Rewrite $\sqrt[3]{\frac{8}{5}}$ as $\frac{\sqrt[3]{8}}{\sqrt[3]{5}}$ .

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

Step 4

Simplify the numerator.

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

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

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

Simplify the numerator.

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

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

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.16960709 \dots$

Simplify the Radical Expression ( cube root of 24)/( cube root of 15)

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