Algebra Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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

Step 3

Combine using the product rule for radicals.

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Simplify each term.

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Step 7.1

Pull terms out from under the radical.

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

Step 7.2

Pull terms out from under the radical.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $h$ by $h$ by adding the exponents.

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Step 7.5.1

Move $h$ .

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

Step 7.5.2

Multiply $h$ by $h$ .

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

Step 8

Reorder factors in $h (x \sqrt{x + h}) + h x \sqrt{x} + h^{2} \sqrt{x}$ .

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