Precalculus Examples

${(\sqrt{a} + \sqrt{b})}^{4}$

Step 1

Use the Binomial Theorem.

${\sqrt{a}}^{4} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{a}}^{4}$ as $a^{2}$ .

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Step 2.1.1.1

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

${(a^{\frac{1}{2}})}^{4} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.2

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

$a^{\frac{1}{2} \cdot 4} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$a^{\frac{4}{2}} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.1.4.1

Factor $2$ out of $4$ .

$a^{\frac{2 \cdot 2}{2}} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.4.2

Cancel the common factors.

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Step 2.1.1.4.2.1

Factor $2$ out of $2$ .

$a^{\frac{2 \cdot 2}{2 (1)}} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.4.2.2

Cancel the common factor.

$a^{\frac{2 \cdot 2}{2 \cdot 1}} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.4.2.3

Rewrite the expression.

$a^{\frac{2}{1}} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.1.4.2.4

Divide $2$ by $1$ .

$a^{2} + 4 {\sqrt{a}}^{3} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.2

Rewrite ${\sqrt{a}}^{3}$ as $\sqrt{a^{3}}$ .

$a^{2} + 4 \sqrt{a^{3}} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.3

Factor out $a^{2}$ .

$a^{2} + 4 \sqrt{a^{2} a} \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.4

Pull terms out from under the radical.

$a^{2} + 4 (a \sqrt{a}) \sqrt{b} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.5

Combine using the product rule for radicals.

$a^{2} + 4 a \sqrt{b a} + 6 {\sqrt{a}}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6

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

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Step 2.1.6.1

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

$a^{2} + 4 a \sqrt{b a} + 6 {(a^{\frac{1}{2}})}^{2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6.2

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

$a^{2} + 4 a \sqrt{b a} + 6 a^{\frac{1}{2} \cdot 2} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6.3

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

$a^{2} + 4 a \sqrt{b a} + 6 a^{\frac{2}{2}} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

$a^{2} + 4 a \sqrt{b a} + 6 a^{\frac{2}{2}} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6.4.2

Rewrite the expression.

$a^{2} + 4 a \sqrt{b a} + 6 a^{1} {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.6.5

Simplify.

$a^{2} + 4 a \sqrt{b a} + 6 a {\sqrt{b}}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7

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

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Step 2.1.7.1

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

$a^{2} + 4 a \sqrt{b a} + 6 a {(b^{\frac{1}{2}})}^{2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7.2

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

$a^{2} + 4 a \sqrt{b a} + 6 a b^{\frac{1}{2} \cdot 2} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7.3

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

$a^{2} + 4 a \sqrt{b a} + 6 a b^{\frac{2}{2}} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$a^{2} + 4 a \sqrt{b a} + 6 a b^{\frac{2}{2}} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7.4.2

Rewrite the expression.

$a^{2} + 4 a \sqrt{b a} + 6 a b^{1} + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.7.5

Simplify.

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 \sqrt{a} {\sqrt{b}}^{3} + {\sqrt{b}}^{4}$

Step 2.1.8

Rewrite ${\sqrt{b}}^{3}$ as $\sqrt{b^{3}}$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 \sqrt{a} \sqrt{b^{3}} + {\sqrt{b}}^{4}$

Step 2.1.9

Factor out $b^{2}$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 \sqrt{a} \sqrt{b^{2} b} + {\sqrt{b}}^{4}$

Step 2.1.10

Pull terms out from under the radical.

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 \sqrt{a} (b \sqrt{b}) + {\sqrt{b}}^{4}$

Step 2.1.11

Combine using the product rule for radicals.

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 (b \sqrt{a b}) + {\sqrt{b}}^{4}$

Step 2.1.12

Rewrite ${\sqrt{b}}^{4}$ as $b^{2}$ .

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Step 2.1.12.1

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

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + {(b^{\frac{1}{2}})}^{4}$

Step 2.1.12.2

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

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{1}{2} \cdot 4}$

Step 2.1.12.3

Combine $\frac{1}{2}$ and $4$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{4}{2}}$

Step 2.1.12.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.12.4.1

Factor $2$ out of $4$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{2 \cdot 2}{2}}$

Step 2.1.12.4.2

Cancel the common factors.

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Step 2.1.12.4.2.1

Factor $2$ out of $2$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{2 \cdot 2}{2 (1)}}$

Step 2.1.12.4.2.2

Cancel the common factor.

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 2.1.12.4.2.3

Rewrite the expression.

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{\frac{2}{1}}$

Step 2.1.12.4.2.4

Divide $2$ by $1$ .

$a^{2} + 4 a \sqrt{b a} + 6 a b + 4 b \sqrt{a b} + b^{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Reorder $b$ and $a$ .

$a^{2} + 4 a \sqrt{a b} + 6 a b + 4 b \sqrt{a b} + b^{2}$

Step 2.2.2

Move $4 b \sqrt{a b}$ .

$a^{2} + 4 a \sqrt{a b} + 6 a b + b^{2} + 4 b \sqrt{a b}$

Step 2.2.3

Move $4 a \sqrt{a b}$ .

$a^{2} + 6 a b + b^{2} + 4 a \sqrt{a b} + 4 b \sqrt{a b}$