Simplify the Radical Expression fourth root of (7x^3)/(4b^2)

$\sqrt[4]{\frac{7 x^{3}}{4 b^{2}}}$

Rewrite $\sqrt[4]{\frac{7 x^{3}}{4 b^{2}}}$ as $\frac{\sqrt[4]{7 x^{3}}}{\sqrt[4]{4 b^{2}}}$ .

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt[4]{4 b^{2}}}$

Simplify the denominator.

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt[4]{{(2 b)}^{2}}}$

Rewrite $\sqrt[4]{{(2 b)}^{2}}$ as $\sqrt{\sqrt{{(2 b)}^{2}}}$ .

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt{\sqrt{{(2 b)}^{2}}}}$

Pull terms out from under the radical.

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt{| 2 b |}}$

Remove non-negative terms from the absolute value.

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt{2 | b |}}$

Multiply $\frac{\sqrt[4]{7 x^{3}}}{\sqrt{2 | b |}}$ by $\frac{\sqrt{2 | b |}}{\sqrt{2 | b |}}$ .

$\frac{\sqrt[4]{7 x^{3}}}{\sqrt{2 | b |}} \cdot \frac{\sqrt{2 | b |}}{\sqrt{2 | b |}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{7 x^{3}}}{\sqrt{2 | b |}}$ by $\frac{\sqrt{2 | b |}}{\sqrt{2 | b |}}$ .

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{\sqrt{2 | b |} \sqrt{2 | b |}}$

Raise $\sqrt{2 | b |}$ to the power of $1$ .

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{\sqrt{2 | b |}}^{1} \sqrt{2 | b |}}$

Raise $\sqrt{2 | b |}$ to the power of $1$ .

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{\sqrt{2 | b |}}^{1} {\sqrt{2 | b |}}^{1}}$

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

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{\sqrt{2 | b |}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{\sqrt{2 | b |}}^{2}}$

Rewrite ${\sqrt{2 | b |}}^{2}$ as $2 | b |$ .

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

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{({(2 | b |)}^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{(2 | b |)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{(2 | b |)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{(2 | b |)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{{(2 | b |)}^{1}}$

Simplify.

$\frac{\sqrt[4]{7 x^{3}} \sqrt{2 | b |}}{2 | b |}$

Simplify the numerator.

Rewrite the expression using the least common index of $4$ .

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

$\frac{\sqrt[4]{7 x^{3}} {(2 | b |)}^{\frac{1}{2}}}{2 | b |}$

Rewrite ${(2 | b |)}^{\frac{1}{2}}$ as ${(2 | b |)}^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{7 x^{3}} {(2 | b |)}^{\frac{2}{4}}}{2 | b |}$

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

$\frac{\sqrt[4]{7 x^{3}} \sqrt[4]{{(2 | b |)}^{2}}}{2 | b |}$

Combine using the product rule for radicals.

$\frac{\sqrt[4]{7 x^{3} {(2 | b |)}^{2}}}{2 | b |}$

Apply the product rule to $2 | b |$ .

$\frac{\sqrt[4]{7 x^{3} \cdot 2^{2} {| b |}^{2}}}{2 | b |}$

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

$\frac{\sqrt[4]{7 x^{3} \cdot 4 {| b |}^{2}}}{2 | b |}$

Remove the absolute value in ${| b |}^{2}$ because exponentiations with even powers are always positive.

$\frac{\sqrt[4]{7 x^{3} \cdot 4 b^{2}}}{2 | b |}$

Multiply $4$ by $7$ .

$\frac{\sqrt[4]{28 x^{3} b^{2}}}{2 | b |}$

Simplify the Radical Expression fourth root of (7x^3)/(4b^2)

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