Simplify fifth root of (( square root of 4x)^2( cube root of 3x)^3)/(( square root of 5)^2)

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

Step 1

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Apply the product rule to $2 \sqrt{x}$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

Combine exponents.

Multiply $3$ by $4$ .

$\sqrt[5]{\frac{12 x \cdot x}{{\sqrt{5}}^{2}}}$

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

$\sqrt[5]{\frac{12 (x^{1} x)}{{\sqrt{5}}^{2}}}$

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

$\sqrt[5]{\frac{12 (x^{1} x^{1})}{{\sqrt{5}}^{2}}}$

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

$\sqrt[5]{\frac{12 x^{1 + 1}}{{\sqrt{5}}^{2}}}$

Add $1$ and $1$ .

$\sqrt[5]{\frac{12 x^{2}}{{\sqrt{5}}^{2}}}$

Step 2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt[5]{\frac{12 x^{2}}{{(5^{\frac{1}{2}})}^{2}}}$

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

$\sqrt[5]{\frac{12 x^{2}}{5^{\frac{1}{2} \cdot 2}}}$

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

$\sqrt[5]{\frac{12 x^{2}}{5^{\frac{2}{2}}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt[5]{\frac{12 x^{2}}{5^{\frac{2}{2}}}}$

Rewrite the expression.

$\sqrt[5]{\frac{12 x^{2}}{5^{1}}}$

Evaluate the exponent.

$\sqrt[5]{\frac{12 x^{2}}{5}}$

Step 3

Rewrite $\sqrt[5]{\frac{12 x^{2}}{5}}$ as $\frac{\sqrt[5]{12 x^{2}}}{\sqrt[5]{5}}$ .

$\frac{\sqrt[5]{12 x^{2}}}{\sqrt[5]{5}}$

Step 4

Multiply $\frac{\sqrt[5]{12 x^{2}}}{\sqrt[5]{5}}$ by $\frac{{\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{4}}$ .

$\frac{\sqrt[5]{12 x^{2}}}{\sqrt[5]{5}} \cdot \frac{{\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{4}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{12 x^{2}}}{\sqrt[5]{5}}$ by $\frac{{\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{4}}$ .

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{\sqrt[5]{5} {\sqrt[5]{5}}^{4}}$

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

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{1} {\sqrt[5]{5}}^{4}}$

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

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{{\sqrt[5]{5}}^{5}}$

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

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

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{{(5^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{5^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{5^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{5^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{5^{1}}$

Evaluate the exponent.

$\frac{\sqrt[5]{12 x^{2}} {\sqrt[5]{5}}^{4}}{5}$

Step 6

Simplify the numerator.

Rewrite ${\sqrt[5]{5}}^{4}$ as ${(5^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{12 x^{2}} \sqrt[5]{5^{4}}}{5}$

Raise $5$ to the power of $4$ .

$\frac{\sqrt[5]{12 x^{2}} \sqrt[5]{625}}{5}$

Step 7

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[5]{12 x^{2} \cdot 625}}{5}$

Multiply $625$ by $12$ .

$\frac{\sqrt[5]{7500 x^{2}}}{5}$

Simplify fifth root of (( square root of 4x)^2( cube root of 3x)^3)/(( square root of 5)^2)

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