Simplify ( cube root of x)/( cube root of 24x)

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

Step 1

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

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

Step 2

Reduce the expression $\frac{x}{24 x}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

$\sqrt[3]{\frac{1}{24}}$

Step 3

Rewrite $\sqrt[3]{\frac{1}{24}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{24}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{24}}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{24}}$

Step 5

Simplify the denominator.

Rewrite $24$ as $2^{3} \cdot 3$ .

Factor $8$ out of $24$ .

$\frac{1}{\sqrt[3]{8 (3)}}$

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

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

Pull terms out from under the radical.

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{3}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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

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

Raise $3$ to the power of $2$ .

$\frac{\sqrt[3]{9}}{2 \cdot 3}$

Step 9

Multiply $2$ by $3$ .

$\frac{\sqrt[3]{9}}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{9}}{6}$

Decimal Form:

$0.34668063 \dots$

Simplify ( cube root of x)/( cube root of 24x)

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