Question

Operations with Algebraic Expressions Perform the indicated operations and simplify.

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

Short answer

The simplified form of the given expression is $\frac{1}{\sqrt{x+h}+\sqrt{x}}$.

Step-by-step solution

Step 1. Rationalizing the Numerator.

If a fraction has numerator of the form $A+B\sqrt{C}$, the multiply numerator and denominator by its conjugate radical$A-B\sqrt{C}$ to rationalize the numerator.

Step 2. Given Information.

The given expression is:

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

Step 3. Simplify the given expression.

The numerator of the given expression is$\sqrt{x+h}-\sqrt{x}$ and its conjugate radical is $\sqrt{x+h}+\sqrt{x}$.

Multiply numerator and denominator of the given expression by $\sqrt{x+h}+\sqrt{x}$.

$\begin{array}{c}\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h}\times \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\\ =\frac{{\left(\sqrt{x+h}\right)}^2-{\left(\sqrt{x}\right)}^2}{h\left(\sqrt{x+h}+\sqrt{x}\right)}\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right]\\ =\frac{x+h-x}{h\left(\sqrt{x+h}+\sqrt{x}\right)}\\ =\frac{h}{h\left(\sqrt{x+h}+\sqrt{x}\right)}\\ =\frac{1}{\sqrt{x+h}+\sqrt{x}}\end{array}$

Thus, the simplified form of the given expression is $\frac{1}{\sqrt{x+h}+\sqrt{x}}$.