Precalculus Examples

$\frac{\frac{1}{\sqrt{4 + h}} - \frac{1}{2}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

To write $\frac{1}{\sqrt{4 + h}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{1}{\sqrt{4 + h}} \cdot \frac{2}{2} - \frac{1}{2}}{h}$

Step 1.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{4 + h}}{\sqrt{4 + h}}$ .

$\frac{\frac{1}{\sqrt{4 + h}} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{\sqrt{4 + h}}{\sqrt{4 + h}}}{h}$

Step 1.3

Write each expression with a common denominator of $\sqrt{4 + h} \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{\sqrt{4 + h}}$ by $\frac{2}{2}$ .

$\frac{\frac{2}{\sqrt{4 + h} \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{4 + h}}{\sqrt{4 + h}}}{h}$

Step 1.3.2

Multiply $\frac{1}{2}$ by $\frac{\sqrt{4 + h}}{\sqrt{4 + h}}$ .

$\frac{\frac{2}{\sqrt{4 + h} \cdot 2} - \frac{\sqrt{4 + h}}{2 \sqrt{4 + h}}}{h}$

Step 1.3.3

Reorder the factors of $\sqrt{4 + h} \cdot 2$ .

$\frac{\frac{2}{2 \sqrt{4 + h}} - \frac{\sqrt{4 + h}}{2 \sqrt{4 + h}}}{h}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{2 - \sqrt{4 + h}}{2 \sqrt{4 + h}}}{h}$

Step 1.5

Multiply $\frac{2 - \sqrt{4 + h}}{2 \sqrt{4 + h}}$ by $\frac{\sqrt{4 + h}}{\sqrt{4 + h}}$ .

$\frac{\frac{2 - \sqrt{4 + h}}{2 \sqrt{4 + h}} \cdot \frac{\sqrt{4 + h}}{\sqrt{4 + h}}}{h}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{2 - \sqrt{4 + h}}{2 \sqrt{4 + h}}$ by $\frac{\sqrt{4 + h}}{\sqrt{4 + h}}$ .

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 \sqrt{4 + h} \sqrt{4 + h}}}{h}$

Step 1.6.2

Move $\sqrt{4 + h}$ .

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 (\sqrt{4 + h} \sqrt{4 + h})}}{h}$

Step 1.6.3

Raise $\sqrt{4 + h}$ to the power of $1$ .

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 ({\sqrt{4 + h}}^{1} \sqrt{4 + h})}}{h}$

Step 1.6.4

Raise $\sqrt{4 + h}$ to the power of $1$ .

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 ({\sqrt{4 + h}}^{1} {\sqrt{4 + h}}^{1})}}{h}$

Step 1.6.5

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

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {\sqrt{4 + h}}^{1 + 1}}}{h}$

Step 1.6.6

Add $1$ and $1$ .

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {\sqrt{4 + h}}^{2}}}{h}$

Step 1.6.7

Rewrite ${\sqrt{4 + h}}^{2}$ as $4 + h$ .

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Step 1.6.7.1

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

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {({(4 + h)}^{\frac{1}{2}})}^{2}}}{h}$

Step 1.6.7.2

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

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {(4 + h)}^{\frac{1}{2} \cdot 2}}}{h}$

Step 1.6.7.3

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

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {(4 + h)}^{\frac{2}{2}}}}{h}$

Step 1.6.7.4

Cancel the common factor of $2$ .

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Step 1.6.7.4.1

Cancel the common factor.

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {(4 + h)}^{\frac{2}{2}}}}{h}$

Step 1.6.7.4.2

Rewrite the expression.

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 {(4 + h)}^{1}}}{h}$

Step 1.6.7.5

Simplify.

$\frac{\frac{(2 - \sqrt{4 + h}) \sqrt{4 + h}}{2 (4 + h)}}{h}$

Step 1.7

Apply the distributive property.

$\frac{\frac{2 \sqrt{4 + h} - \sqrt{4 + h} \sqrt{4 + h}}{2 (4 + h)}}{h}$

Step 1.8

Multiply $- \sqrt{4 + h} \sqrt{4 + h}$ .

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Step 1.8.1

Raise $\sqrt{4 + h}$ to the power of $1$ .

$\frac{\frac{2 \sqrt{4 + h} - ({\sqrt{4 + h}}^{1} \sqrt{4 + h})}{2 (4 + h)}}{h}$

Step 1.8.2

Raise $\sqrt{4 + h}$ to the power of $1$ .

$\frac{\frac{2 \sqrt{4 + h} - ({\sqrt{4 + h}}^{1} {\sqrt{4 + h}}^{1})}{2 (4 + h)}}{h}$

Step 1.8.3

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

$\frac{\frac{2 \sqrt{4 + h} - {\sqrt{4 + h}}^{1 + 1}}{2 (4 + h)}}{h}$

Step 1.8.4

Add $1$ and $1$ .

$\frac{\frac{2 \sqrt{4 + h} - {\sqrt{4 + h}}^{2}}{2 (4 + h)}}{h}$

Step 1.9

Simplify each term.

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Step 1.9.1

Rewrite ${\sqrt{4 + h}}^{2}$ as $4 + h$ .

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Step 1.9.1.1

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

$\frac{\frac{2 \sqrt{4 + h} - {({(4 + h)}^{\frac{1}{2}})}^{2}}{2 (4 + h)}}{h}$

Step 1.9.1.2

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

$\frac{\frac{2 \sqrt{4 + h} - {(4 + h)}^{\frac{1}{2} \cdot 2}}{2 (4 + h)}}{h}$

Step 1.9.1.3

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

$\frac{\frac{2 \sqrt{4 + h} - {(4 + h)}^{\frac{2}{2}}}{2 (4 + h)}}{h}$

Step 1.9.1.4

Cancel the common factor of $2$ .

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Step 1.9.1.4.1

Cancel the common factor.

$\frac{\frac{2 \sqrt{4 + h} - {(4 + h)}^{\frac{2}{2}}}{2 (4 + h)}}{h}$

Step 1.9.1.4.2

Rewrite the expression.

$\frac{\frac{2 \sqrt{4 + h} - {(4 + h)}^{1}}{2 (4 + h)}}{h}$

Step 1.9.1.5

Simplify.

$\frac{\frac{2 \sqrt{4 + h} - (4 + h)}{2 (4 + h)}}{h}$

Step 1.9.2

Apply the distributive property.

$\frac{\frac{2 \sqrt{4 + h} - 1 \cdot 4 - h}{2 (4 + h)}}{h}$

Step 1.9.3

Multiply $- 1$ by $4$ .

$\frac{\frac{2 \sqrt{4 + h} - 4 - h}{2 (4 + h)}}{h}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \sqrt{4 + h} - 4 - h}{2 (4 + h)} \cdot \frac{1}{h}$

Step 3

Multiply $\frac{2 \sqrt{4 + h} - 4 - h}{2 (4 + h)}$ by $\frac{1}{h}$ .

$\frac{2 \sqrt{4 + h} - 4 - h}{2 (4 + h) h}$

Step 4

Reorder factors in $\frac{2 \sqrt{4 + h} - 4 - h}{2 (4 + h) h}$ .

$\frac{2 \sqrt{4 + h} - 4 - h}{2 h (4 + h)}$