Simplify ( cube root of x* square root of x^5)/( square root of 25x^16)

$\frac{\sqrt[3]{x} \cdot \sqrt{x^{5}}}{\sqrt{25 x^{16}}}$

Step 1

Combine $\sqrt{x^{5}}$ and $\sqrt{25 x^{16}}$ into a single radical.

$\sqrt[3]{x} \cdot \sqrt{\frac{x^{5}}{25 x^{16}}}$

Step 2

Reduce the expression $\frac{x^{5}}{25 x^{16}}$ by cancelling the common factors.

Multiply by $1$ .

$\sqrt[3]{x} \cdot \sqrt{\frac{x^{5} \cdot 1}{25 x^{16}}}$

Factor $x^{5}$ out of $25 x^{16}$ .

$\sqrt[3]{x} \cdot \sqrt{\frac{x^{5} \cdot 1}{x^{5} (25 x^{11})}}$

Cancel the common factor.

$\sqrt[3]{x} \cdot \sqrt{\frac{x^{5} \cdot 1}{x^{5} (25 x^{11})}}$

Rewrite the expression.

$\sqrt[3]{x} \cdot \sqrt{\frac{1}{25 x^{11}}}$

Step 3

Rewrite $\frac{1}{25 x^{11}}$ as ${(\frac{1}{5 x^{5}})}^{2} \frac{1}{x}$ .

Factor the perfect power $1^{2}$ out of $1$ .

$\sqrt[3]{x} \cdot \sqrt{\frac{1^{2} \cdot 1}{25 x^{11}}}$

Factor the perfect power ${(5 x^{5})}^{2}$ out of $25 x^{11}$ .

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

Rearrange the fraction $\frac{1^{2} \cdot 1}{{(5 x^{5})}^{2} x}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\sqrt{\frac{1}{x}}$ as $\frac{\sqrt{1}}{\sqrt{x}}$ .

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

Step 6

Combine.

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

Step 7

Multiply $\sqrt{1}$ by $1$ .

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

Step 8

Any root of $1$ is $1$ .

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

Step 9

Multiply $\frac{1}{5 x^{5} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 10

Combine and simplify the denominator.

Multiply $\frac{1}{5 x^{5} \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Move $\sqrt{x}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 11

Simplify the denominator.

Raise $x$ to the power of $1$ .

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

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

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

Add $1$ and $5$ .

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

Step 12

Multiply $\sqrt[3]{x} \frac{\sqrt{x}}{5 x^{6}}$ .

Combine $\sqrt[3]{x}$ and $\frac{\sqrt{x}}{5 x^{6}}$ .

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

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

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

$\frac{x^{\frac{1}{3}} \sqrt{x}}{5 x^{6}}$

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

$\frac{x^{\frac{2}{6}} \sqrt{x}}{5 x^{6}}$

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

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

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

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

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

$\frac{\sqrt[6]{x^{2}} x^{\frac{3}{6}}}{5 x^{6}}$

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

$\frac{\sqrt[6]{x^{2}} \sqrt[6]{x^{3}}}{5 x^{6}}$

Combine using the product rule for radicals.

$\frac{\sqrt[6]{x^{2} x^{3}}}{5 x^{6}}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

$\frac{\sqrt[6]{x^{2 + 3}}}{5 x^{6}}$

Add $2$ and $3$ .

$\frac{\sqrt[6]{x^{5}}}{5 x^{6}}$

Simplify ( cube root of x* square root of x^5)/( square root of 25x^16)

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