Precalculus Examples

(\sqrt{h^{2} + 1} + 1) (\sqrt{h^{2} + 1} - 1)

$(\sqrt{h^{2} + 1} + 1) (\sqrt{h^{2} + 1} - 1)$

Step 1

Expand

(\sqrt{h^{2} + 1} + 1) (\sqrt{h^{2} + 1} - 1)

$(\sqrt{h^{2} + 1} + 1) (\sqrt{h^{2} + 1} - 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

\sqrt{h^{2} + 1} (\sqrt{h^{2} + 1} - 1) + 1 (\sqrt{h^{2} + 1} - 1)

$\sqrt{h^{2} + 1} (\sqrt{h^{2} + 1} - 1) + 1 (\sqrt{h^{2} + 1} - 1)$

Step 1.2

Apply the distributive property.

\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 (\sqrt{h^{2} + 1} - 1)

$\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 (\sqrt{h^{2} + 1} - 1)$

Step 1.3

Apply the distributive property.

\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

$\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

$\sqrt{h^{2} + 1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply

\sqrt{h^{2} + 1} \sqrt{h^{2} + 1}

$\sqrt{h^{2} + 1} \sqrt{h^{2} + 1}$ .

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Step 2.1.1.1

Raise

\sqrt{h^{2} + 1}

$\sqrt{h^{2} + 1}$ to the power of

1

$1$ .

{\sqrt{h^{2} + 1}}^{1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${\sqrt{h^{2} + 1}}^{1} \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.1.2

Raise

\sqrt{h^{2} + 1}

$\sqrt{h^{2} + 1}$ to the power of

1

$1$ .

{\sqrt{h^{2} + 1}}^{1} {\sqrt{h^{2} + 1}}^{1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${\sqrt{h^{2} + 1}}^{1} {\sqrt{h^{2} + 1}}^{1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

{\sqrt{h^{2} + 1}}^{1 + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${\sqrt{h^{2} + 1}}^{1 + 1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.1.4

Add

1

$1$ and

1

$1$ .

{\sqrt{h^{2} + 1}}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${\sqrt{h^{2} + 1}}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

{\sqrt{h^{2} + 1}}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${\sqrt{h^{2} + 1}}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2

Rewrite

{\sqrt{h^{2} + 1}}^{2}

${\sqrt{h^{2} + 1}}^{2}$ as

h^{2} + 1

$h^{2} + 1$ .

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Step 2.1.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{h^{2} + 1}

$\sqrt{h^{2} + 1}$ as

{(h^{2} + 1)}^{\frac{1}{2}}

${(h^{2} + 1)}^{\frac{1}{2}}$ .

{({(h^{2} + 1)}^{\frac{1}{2}})}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${({(h^{2} + 1)}^{\frac{1}{2}})}^{2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{(h^{2} + 1)}^{\frac{1}{2} \cdot 2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${(h^{2} + 1)}^{\frac{1}{2} \cdot 2} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

{(h^{2} + 1)}^{\frac{2}{2}} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1

${(h^{2} + 1)}^{\frac{2}{2}} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2.4

Cancel the common factor of

2

$2$ .

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Step 2.1.2.4.1

Cancel the common factor.

${(h^{2} + 1)}^{\frac{2}{2}} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2.4.2

Rewrite the expression.

${(h^{2} + 1)}^{1} + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.2.5

Simplify.

$h^{2} + 1 + \sqrt{h^{2} + 1} \cdot - 1 + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.3

Move

$- 1$ to the left of

$\sqrt{h^{2} + 1}$ .

$h^{2} + 1 - 1 \cdot \sqrt{h^{2} + 1} + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.4

Rewrite

$- 1 \sqrt{h^{2} + 1}$ as

$- \sqrt{h^{2} + 1}$ .

$h^{2} + 1 - \sqrt{h^{2} + 1} + 1 \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.5

Multiply

$\sqrt{h^{2} + 1}$ by

$1$ .

$h^{2} + 1 - \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} + 1 \cdot - 1$

Step 2.1.6

Multiply

$- 1$ by

$1$ .

$h^{2} + 1 - \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1} - 1$

Step 2.2

Subtract

$1$ from

$1$ .

$h^{2} + 0 - \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1}$

Step 2.3

Add

$h^{2}$ and

$0$ .

$h^{2} - \sqrt{h^{2} + 1} + \sqrt{h^{2} + 1}$

Step 2.4

Add

$- \sqrt{h^{2} + 1}$ and

$\sqrt{h^{2} + 1}$ .

$h^{2} + 0$

Step 2.5

Add

$h^{2}$ and

$0$ .

$h^{2}$