Precalculus Examples

$(\sqrt{a} - \frac{1}{b}) (\sqrt{a} + \frac{1}{b})$

Step 1

Expand $(\sqrt{a} - \frac{1}{b}) (\sqrt{a} + \frac{1}{b})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{a} (\sqrt{a} + \frac{1}{b}) - \frac{1}{b} (\sqrt{a} + \frac{1}{b})$

Step 1.2

Apply the distributive property.

$\sqrt{a} \sqrt{a} + \sqrt{a} \frac{1}{b} - \frac{1}{b} (\sqrt{a} + \frac{1}{b})$

Step 1.3

Apply the distributive property.

$\sqrt{a} \sqrt{a} + \sqrt{a} \frac{1}{b} - \frac{1}{b} \sqrt{a} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{a} \sqrt{a} + \sqrt{a} \frac{1}{b} - \frac{1}{b} \sqrt{a} - \frac{1}{b} \cdot \frac{1}{b}$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{a} \frac{1}{b}$ and $- \frac{1}{b} \sqrt{a}$ .

$\sqrt{a} \sqrt{a} + \sqrt{a} \frac{1}{b} - \sqrt{a} \frac{1}{b} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.1.2

Subtract $\sqrt{a} \frac{1}{b}$ from $\sqrt{a} \frac{1}{b}$ .

$\sqrt{a} \sqrt{a} + 0 - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.1.3

Add $\sqrt{a} \sqrt{a}$ and $0$ .

$\sqrt{a} \sqrt{a} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\sqrt{a} \sqrt{a}$ .

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Step 2.2.1.1

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

${\sqrt{a}}^{1} \sqrt{a} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.1.2

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

${\sqrt{a}}^{1} {\sqrt{a}}^{1} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.1.3

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

${\sqrt{a}}^{1 + 1} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.1.4

Add $1$ and $1$ .

${\sqrt{a}}^{2} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

$a^{1} - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.2.5

Simplify.

$a - \frac{1}{b} \cdot \frac{1}{b}$

Step 2.2.3

Multiply $- \frac{1}{b} \cdot \frac{1}{b}$ .

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Step 2.2.3.1

Multiply $\frac{1}{b}$ by $\frac{1}{b}$ .

$a - \frac{1}{b \cdot b}$

Step 2.2.3.2

Raise $b$ to the power of $1$ .

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

Step 2.2.3.3

Raise $b$ to the power of $1$ .

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

Step 2.2.3.4

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

$a - \frac{1}{b^{1 + 1}}$

Step 2.2.3.5

Add $1$ and $1$ .

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