Question

rationalize the numerator. assume that all variables represent positive real numbers.\\(\frac{sqrt{x + 5}-sqrt{x}}{5}\\)\\(\frac{sqrt{x + 5}-sqrt{x}}{5}=square\\) (simplify your answer.)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the numerator.

The conjugate of $\sqrt{x + 5}-\sqrt{x}$ is $\sqrt{x + 5}+\sqrt{x}$. So we multiply the fraction by $\frac{\sqrt{x + 5}+\sqrt{x}}{\sqrt{x + 5}+\sqrt{x}}$:
$$\frac{\sqrt{x + 5}-\sqrt{x}}{5}\times\frac{\sqrt{x + 5}+\sqrt{x}}{\sqrt{x + 5}+\sqrt{x}}=\frac{(\sqrt{x + 5}-\sqrt{x})(\sqrt{x + 5}+\sqrt{x})}{5(\sqrt{x + 5}+\sqrt{x})}$$

Step2: Apply the difference of squares formula $(a - b)(a + b)=a^2 - b^2$ to the numerator.

Here, $a=\sqrt{x + 5}$ and $b = \sqrt{x}$. So the numerator becomes $(\sqrt{x + 5})^2-(\sqrt{x})^2=(x + 5)-x$.
$$\frac{(x + 5)-x}{5(\sqrt{x + 5}+\sqrt{x})}$$

Step3: Simplify the numerator.

Simplify $(x + 5)-x$: $x+5 - x=5$.
$$\frac{5}{5(\sqrt{x + 5}+\sqrt{x})}$$

Step4: Cancel out the common factor of 5.

Cancel the 5 in the numerator and the 5 in the denominator:
$$\frac{1}{\sqrt{x + 5}+\sqrt{x}}$$

Answer:

$\frac{1}{\sqrt{x + 5}+\sqrt{x}}$