Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

Apply the product rule to $2 b$ .

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

Step 5.4

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

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

Step 5.5

Multiply $4$ by $7$ .

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