Simplify cube root of (125x^11)/4

$\sqrt[3]{\frac{125 x^{11}}{4}}$

Step 1

Rewrite $\sqrt[3]{\frac{125 x^{11}}{4}}$ as $\frac{\sqrt[3]{125 x^{11}}}{\sqrt[3]{4}}$ .

$\frac{\sqrt[3]{125 x^{11}}}{\sqrt[3]{4}}$

Step 2

Simplify the numerator.

Rewrite $125 x^{11}$ as ${(5 x^{3})}^{3} x^{2}$ .

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

$\frac{\sqrt[3]{5^{3} x^{11}}}{\sqrt[3]{4}}$

Factor out $x^{9}$ .

$\frac{\sqrt[3]{5^{3} (x^{9} x^{2})}}{\sqrt[3]{4}}$

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

$\frac{\sqrt[3]{5^{3} ({(x^{3})}^{3} x^{2})}}{\sqrt[3]{4}}$

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

$\frac{\sqrt[3]{{(5 x^{3})}^{3} x^{2}}}{\sqrt[3]{4}}$

Pull terms out from under the radical.

$\frac{5 x^{3} \sqrt[3]{x^{2}}}{\sqrt[3]{4}}$

Step 3

Multiply $\frac{5 x^{3} \sqrt[3]{x^{2}}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{5 x^{3} \sqrt[3]{x^{2}}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Raise $\sqrt[3]{4}$ to the power of $1$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

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

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

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

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

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

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{4^{1}}$

Evaluate the exponent.

$\frac{5 x^{3} \sqrt[3]{x^{2}} {\sqrt[3]{4}}^{2}}{4}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{4}}^{2}$ as ${(4^{2})}^{\frac{1}{3}}$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} \sqrt[3]{4^{2}}}{4}$

Raise $4$ to the power of $2$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} \sqrt[3]{16}}{4}$

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$\frac{5 x^{3} \sqrt[3]{x^{2}} \sqrt[3]{8 (2)}}{4}$

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

$\frac{5 x^{3} \sqrt[3]{x^{2}} \sqrt[3]{2^{3} \cdot 2}}{4}$

Pull terms out from under the radical.

$\frac{5 x^{3} \sqrt[3]{x^{2}} \cdot 2 \sqrt[3]{2}}{4}$

Combine exponents.

Multiply $2$ by $5$ .

$\frac{10 x^{3} \sqrt[3]{x^{2}} \sqrt[3]{2}}{4}$

Combine using the product rule for radicals.

$\frac{10 x^{3} \sqrt[3]{2 x^{2}}}{4}$

Step 6

Cancel the common factor of $10$ and $4$ .

Factor $2$ out of $10 x^{3} \sqrt[3]{2 x^{2}}$ .

$\frac{2 (5 x^{3} \sqrt[3]{2 x^{2}})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify cube root of (125x^11)/4

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