Basic Math Examples

$\frac{\frac{1}{\sqrt{z + h}} - \frac{1}{z}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\frac{1}{\sqrt{z + h}} \cdot \frac{z}{z} - \frac{1}{z}}{h}$

Step 1.2

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

$\frac{\frac{1}{\sqrt{z + h}} \cdot \frac{z}{z} - \frac{1}{z} \cdot \frac{\sqrt{z + h}}{\sqrt{z + h}}}{h}$

Step 1.3

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

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Step 1.3.1

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

$\frac{\frac{z}{\sqrt{z + h} z} - \frac{1}{z} \cdot \frac{\sqrt{z + h}}{\sqrt{z + h}}}{h}$

Step 1.3.2

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

$\frac{\frac{z}{\sqrt{z + h} z} - \frac{\sqrt{z + h}}{z \sqrt{z + h}}}{h}$

Step 1.3.3

Reorder the factors of $\sqrt{z + h} z$ .

$\frac{\frac{z}{z \sqrt{z + h}} - \frac{\sqrt{z + h}}{z \sqrt{z + h}}}{h}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{z - \sqrt{z + h}}{z \sqrt{z + h}}}{h}$

Step 1.5

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

$\frac{\frac{z - \sqrt{z + h}}{z \sqrt{z + h}} \cdot \frac{\sqrt{z + h}}{\sqrt{z + h}}}{h}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$\frac{\frac{(z - \sqrt{z + h}) \sqrt{z + h}}{z \sqrt{z + h} \sqrt{z + h}}}{h}$

Step 1.6.2

Move $\sqrt{z + h}$ .

$\frac{\frac{(z - \sqrt{z + h}) \sqrt{z + h}}{z (\sqrt{z + h} \sqrt{z + h})}}{h}$

Step 1.6.3

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

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

Step 1.6.4

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

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

Step 1.6.5

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

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

Step 1.6.6

Add $1$ and $1$ .

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

Step 1.6.7

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

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Step 1.6.7.1

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

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

Step 1.6.7.2

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

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

Step 1.6.7.3

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

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

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.

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

Step 1.6.7.4.2

Rewrite the expression.

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

Step 1.6.7.5

Simplify.

$\frac{\frac{(z - \sqrt{z + h}) \sqrt{z + h}}{z (z + h)}}{h}$

Step 1.7

Apply the distributive property.

$\frac{\frac{z \sqrt{z + h} - \sqrt{z + h} \sqrt{z + h}}{z (z + h)}}{h}$

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Add $1$ and $1$ .

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

Step 1.9

Simplify each term.

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Step 1.9.1

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

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Step 1.9.1.1

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

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

Step 1.9.1.2

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

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

Step 1.9.1.3

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

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

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.

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

Step 1.9.1.4.2

Rewrite the expression.

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

Step 1.9.1.5

Simplify.

$\frac{\frac{z \sqrt{z + h} - (z + h)}{z (z + h)}}{h}$

Step 1.9.2

Apply the distributive property.

$\frac{\frac{z \sqrt{z + h} - z - h}{z (z + h)}}{h}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

$\frac{z \sqrt{z + h} - z - h}{z (z + h) h}$

Step 4

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

$\frac{z \sqrt{z + h} - z - h}{z h (z + h)}$