Algebra Examples

$\frac{\sqrt[5]{{(a + b)}^{4}}}{\sqrt{a + b}}$

Step 1

Multiply to rationalize the numerator.

$\frac{\sqrt[5]{{(a + b)}^{4}} {\sqrt[5]{{(a + b)}^{4}}}^{4}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2

Simplify.

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Step 2.1

Raise $\sqrt[5]{{(a + b)}^{4}}$ to the power of $1$ .

$\frac{{\sqrt[5]{{(a + b)}^{4}}}^{1} {\sqrt[5]{{(a + b)}^{4}}}^{4}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.2

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

$\frac{{\sqrt[5]{{(a + b)}^{4}}}^{1 + 4}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.3

Add $1$ and $4$ .

$\frac{{\sqrt[5]{{(a + b)}^{4}}}^{5}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4

Rewrite ${\sqrt[5]{{(a + b)}^{4}}}^{5}$ as ${(a + b)}^{4}$ .

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(a + b)}^{4}}$ as ${(a + b)}^{\frac{4}{5}}$ .

$\frac{{({(a + b)}^{\frac{4}{5}})}^{5}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.2

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

$\frac{{(a + b)}^{\frac{4}{5} \cdot 5}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.3

Combine $\frac{4}{5}$ and $5$ .

$\frac{{(a + b)}^{\frac{4 \cdot 5}{5}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.4

Multiply $4$ by $5$ .

$\frac{{(a + b)}^{\frac{20}{5}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.5

Cancel the common factor of $20$ and $5$ .

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Step 2.4.5.1

Factor $5$ out of $20$ .

$\frac{{(a + b)}^{\frac{5 \cdot 4}{5}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.5.2

Cancel the common factors.

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Step 2.4.5.2.1

Factor $5$ out of $5$ .

$\frac{{(a + b)}^{\frac{5 \cdot 4}{5 (1)}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.5.2.2

Cancel the common factor.

$\frac{{(a + b)}^{\frac{5 \cdot 4}{5 \cdot 1}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.5.2.3

Rewrite the expression.

$\frac{{(a + b)}^{\frac{4}{1}}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.4.5.2.4

Divide $4$ by $1$ .

$\frac{{(a + b)}^{4}}{\sqrt{a + b} {\sqrt[5]{{(a + b)}^{4}}}^{4}}$

Step 2.5

Start simplifying.

$\frac{{(a + b)}^{4}}{\sqrt{a + b} \sqrt[5]{{({(a + b)}^{4})}^{4}}}$

Step 2.6

Multiply the exponents in ${({(a + b)}^{4})}^{4}$ .

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Step 2.6.1

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

$\frac{{(a + b)}^{4}}{\sqrt{a + b} \sqrt[5]{{(a + b)}^{4 \cdot 4}}}$

Step 2.6.2

Multiply $4$ by $4$ .

$\frac{{(a + b)}^{4}}{\sqrt{a + b} \sqrt[5]{{(a + b)}^{16}}}$

Step 2.7

Rewrite ${(a + b)}^{16}$ as ${({(a + b)}^{3})}^{5} (a + b)$ .

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Step 2.7.1

Factor out ${(a + b)}^{15}$ .

$\frac{{(a + b)}^{4}}{\sqrt{a + b} \sqrt[5]{{(a + b)}^{15} (a + b)}}$

Step 2.7.2

Rewrite ${(a + b)}^{15}$ as ${({(a + b)}^{3})}^{5}$ .

$\frac{{(a + b)}^{4}}{\sqrt{a + b} \sqrt[5]{{({(a + b)}^{3})}^{5} (a + b)}}$

Step 2.8

Pull terms out from under the radical.

$\frac{{(a + b)}^{4}}{\sqrt{a + b} ({(a + b)}^{3} \sqrt[5]{a + b})}$

Step 2.9

Rewrite the expression using the least common index of $10$ .

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Step 2.9.1

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

$\frac{{(a + b)}^{4}}{{(a + b)}^{\frac{1}{5}} \sqrt{a + b} {(a + b)}^{3}}$

Step 2.9.2

Rewrite ${(a + b)}^{\frac{1}{5}}$ as ${(a + b)}^{\frac{2}{10}}$ .

$\frac{{(a + b)}^{4}}{{(a + b)}^{\frac{2}{10}} \sqrt{a + b} {(a + b)}^{3}}$

Step 2.9.3

Rewrite ${(a + b)}^{\frac{2}{10}}$ as $\sqrt[10]{{(a + b)}^{2}}$ .

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2}} \sqrt{a + b} {(a + b)}^{3}}$

Step 2.9.4

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

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2}} {(a + b)}^{\frac{1}{2}} {(a + b)}^{3}}$

Step 2.9.5

Rewrite ${(a + b)}^{\frac{1}{2}}$ as ${(a + b)}^{\frac{5}{10}}$ .

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2}} {(a + b)}^{\frac{5}{10}} {(a + b)}^{3}}$

Step 2.9.6

Rewrite ${(a + b)}^{\frac{5}{10}}$ as $\sqrt[10]{{(a + b)}^{5}}$ .

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2}} \sqrt[10]{{(a + b)}^{5}} {(a + b)}^{3}}$

Step 2.10

Combine using the product rule for radicals.

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2} {(a + b)}^{5}} {(a + b)}^{3}}$

Step 2.11

Multiply ${(a + b)}^{2}$ by ${(a + b)}^{5}$ by adding the exponents.

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Step 2.11.1

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

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{2 + 5}} {(a + b)}^{3}}$

Step 2.11.2

Add $2$ and $5$ .

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} {(a + b)}^{3}}$

Step 3

Use the Binomial Theorem.

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3})}$

Step 4

Apply the distributive property.

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} a^{3} + \sqrt[10]{{(a + b)}^{7}} (3 a^{2} b) + \sqrt[10]{{(a + b)}^{7}} (3 a b^{2}) + \sqrt[10]{{(a + b)}^{7}} b^{3}}$

Step 5

Simplify.

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Step 5.1

Rewrite using the commutative property of multiplication.

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} a^{3} + 3 \sqrt[10]{{(a + b)}^{7}} (a^{2} b) + \sqrt[10]{{(a + b)}^{7}} (3 a b^{2}) + \sqrt[10]{{(a + b)}^{7}} b^{3}}$

Step 5.2

Rewrite using the commutative property of multiplication.

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} a^{3} + 3 \sqrt[10]{{(a + b)}^{7}} (a^{2} b) + 3 \sqrt[10]{{(a + b)}^{7}} (a b^{2}) + \sqrt[10]{{(a + b)}^{7}} b^{3}}$

$\frac{{(a + b)}^{4}}{\sqrt[10]{{(a + b)}^{7}} a^{3} + 3 \sqrt[10]{{(a + b)}^{7}} a^{2} b + 3 \sqrt[10]{{(a + b)}^{7}} a b^{2} + \sqrt[10]{{(a + b)}^{7}} b^{3}}$