Algebra Examples

$\frac{a - b}{\sqrt[4]{a^{3}} - \sqrt[4]{a^{2} b}}$

Step 1

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

$\frac{a - b}{a^{\frac{3}{4}} - \sqrt[4]{a^{2} b}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{a^{2} b}$ as ${(a^{2} b)}^{\frac{1}{4}}$ .

$\frac{a - b}{a^{\frac{3}{4}} - {(a^{2} b)}^{\frac{1}{4}}}$

Step 3

Apply the product rule to $a^{2} b$ .

$\frac{a - b}{a^{\frac{3}{4}} - ({(a^{2})}^{\frac{1}{4}} b^{\frac{1}{4}})}$

Step 4

Multiply the exponents in ${(a^{2})}^{\frac{1}{4}}$ .

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Step 4.1

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

$\frac{a - b}{a^{\frac{3}{4}} - (a^{2 (\frac{1}{4})} b^{\frac{1}{4}})}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $4$ .

$\frac{a - b}{a^{\frac{3}{4}} - (a^{2 \frac{1}{2 (2)}} b^{\frac{1}{4}})}$

Step 4.2.2

Cancel the common factor.

$\frac{a - b}{a^{\frac{3}{4}} - (a^{2 \frac{1}{2 \cdot 2}} b^{\frac{1}{4}})}$

Step 4.2.3

Rewrite the expression.

$\frac{a - b}{a^{\frac{3}{4}} - (a^{\frac{1}{2}} b^{\frac{1}{4}})}$

Step 5

Rewrite $a^{\frac{3}{4}} - a^{\frac{1}{2}} b^{\frac{1}{4}}$ in a factored form.

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Step 5.1

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{3}{4}} - a^{\frac{1}{2}} b^{\frac{1}{4}}$ .

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Step 5.1.1

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{3}{4}}$ .

$\frac{a - b}{a^{\frac{1}{2}} a^{\frac{1}{4}} - a^{\frac{1}{2}} b^{\frac{1}{4}}}$

Step 5.1.2

Factor $a^{\frac{1}{2}}$ out of $- a^{\frac{1}{2}} b^{\frac{1}{4}}$ .

$\frac{a - b}{a^{\frac{1}{2}} a^{\frac{1}{4}} + a^{\frac{1}{2}} (- 1 b^{\frac{1}{4}})}$

Step 5.1.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} a^{\frac{1}{4}} + a^{\frac{1}{2}} (- 1 b^{\frac{1}{4}})$ .

$\frac{a - b}{a^{\frac{1}{2}} (a^{\frac{1}{4}} - 1 b^{\frac{1}{4}})}$

Step 5.2

Rewrite $- 1 b^{\frac{1}{4}}$ as $- b^{\frac{1}{4}}$ .

$\frac{a - b}{a^{\frac{1}{2}} (a^{\frac{1}{4}} - b^{\frac{1}{4}})}$