Algebra Examples

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

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

Step 1

Multiply to rationalize the numerator.

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

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

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

Step 2.2

Simplify.

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Step 2.2.1

Subtract

x

$x$ from

x

$x$ .

\frac{h + 0}{h (\sqrt{x + h} + \sqrt{x})}

$\frac{h + 0}{h (\sqrt{x + h} + \sqrt{x})}$

Step 2.2.2

Add

h

$h$ and

0

$0$ .

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

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

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

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

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

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

Step 3

Apply the distributive property.

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

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