Simplify ( cube root of 8x^5)/( square root of x)

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

Step 1

Simplify the numerator.

Rewrite $8 x^{5}$ as ${(2 x)}^{3} x^{2}$ .

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

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

Factor out $x^{3}$ .

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

Rewrite $2^{3} x^{3}$ as ${(2 x)}^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $2 \sqrt[3]{x^{2}} \sqrt{x}$ by $1$ .

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

Step 5

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

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

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

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

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

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

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

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

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

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

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

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

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

$2 (\sqrt[6]{x^{3}} \sqrt[6]{x^{4}})$

Combine using the product rule for radicals.

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

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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

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

Add $3$ and $4$ .

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

Step 6

Factor out $x^{6}$ .

$2 \sqrt[6]{x^{6} x}$

Step 7

Pull terms out from under the radical.

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

Simplify ( cube root of 8x^5)/( square root of x)

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