Algebra Examples

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

Step 1

Combine using the product rule for radicals.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Move $\sqrt{(x + h) x}$ .

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

Step 3.3

Raise $\sqrt{(x + h) x}$ to the power of $1$ .

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

Step 3.4

Raise $\sqrt{(x + h) x}$ to the power of $1$ .

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

Rewrite ${\sqrt{(x + h) x}}^{2}$ as $(x + h) x$ .

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Simplify.

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

Step 4

Simplify the denominator.

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Step 4.1

Rewrite.

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

Step 4.2

Move $\sqrt{(x + h) x}$ .

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

Step 4.3

Raise $\sqrt{(x + h) x}$ to the power of $1$ .

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

Step 4.4

Raise $\sqrt{(x + h) x}$ to the power of $1$ .

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Rewrite ${\sqrt{(x + h) x}}^{2}$ as $(x + h) x$ .

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

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

Step 4.7.4.2

Rewrite the expression.

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

Step 4.7.5

Simplify.

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

Step 4.8

Remove unnecessary parentheses.

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

Step 5

Apply the distributive property.

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

Step 6

Multiply $\sqrt{x} \sqrt{(x + h) x}$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Multiply $- \sqrt{x + h} \sqrt{(x + h) x}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Raise $x + h$ to the power of $1$ .

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

Step 7.3

Raise $x + h$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

Simplify each term.

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Step 8.1

Pull terms out from under the radical.

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

Step 8.2

Reorder $x$ and ${(x + h)}^{2}$ .

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

Step 8.3

Pull terms out from under the radical.

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

Step 8.4

Apply the distributive property.

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

Step 8.5

Apply the distributive property.

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

Step 9

Simplify the numerator.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Factor $x^{\frac{1}{2}}$ out of $x {(x + h)}^{\frac{1}{2}} - x \cdot x^{\frac{1}{2}} - h x^{\frac{1}{2}}$ .

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Step 9.4.1

Reorder $x$ and $h$ .

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

Step 9.4.2

Factor $x^{\frac{1}{2}}$ out of $x {(h + x)}^{\frac{1}{2}}$ .

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

Step 9.4.3

Factor $x^{\frac{1}{2}}$ out of $- x \cdot x^{\frac{1}{2}}$ .

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

Step 9.4.4

Factor $x^{\frac{1}{2}}$ out of $- h x^{\frac{1}{2}}$ .

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

Step 9.4.5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (x^{\frac{1}{2}} {(h + x)}^{\frac{1}{2}}) + x^{\frac{1}{2}} (- x^{\frac{1}{2}} x^{\frac{1}{2}})$ .

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

Step 9.4.6

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (x^{\frac{1}{2}} {(h + x)}^{\frac{1}{2}} - x^{\frac{1}{2}} x^{\frac{1}{2}}) + x^{\frac{1}{2}} (- h)$ .

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

Step 9.5

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 9.5.1

Move $x^{\frac{1}{2}}$ .

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

Step 9.5.2

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

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

Step 9.5.3

Combine the numerators over the common denominator.

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

Step 9.5.4

Add $1$ and $1$ .

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

Step 9.5.5

Divide $2$ by $2$ .

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

Step 9.6

Simplify $- x^{1}$ .

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

Step 10

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 11

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 11.1

Move $x^{- \frac{1}{2}}$ .

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

Step 11.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

Write $1$ as a fraction with a common denominator.

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Add $- 1$ and $2$ .

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

Step 12

Reorder factors in $\frac{x^{\frac{1}{2}} {(h + x)}^{\frac{1}{2}} - x - h}{h (x + h) x^{\frac{1}{2}}}$ .

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