Simplify ( square root of 5+x)/( square root of 5+ square root of x)

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

Step 1

Multiply $\frac{\sqrt{5 + x}}{\sqrt{5} + \sqrt{x}}$ by $\frac{\sqrt{5} - \sqrt{x}}{\sqrt{5} - \sqrt{x}}$ .

$\frac{\sqrt{5 + x}}{\sqrt{5} + \sqrt{x}} \cdot \frac{\sqrt{5} - \sqrt{x}}{\sqrt{5} - \sqrt{x}}$

Step 2

Multiply $\frac{\sqrt{5 + x}}{\sqrt{5} + \sqrt{x}}$ by $\frac{\sqrt{5} - \sqrt{x}}{\sqrt{5} - \sqrt{x}}$ .

$\frac{\sqrt{5 + x} (\sqrt{5} - \sqrt{x})}{(\sqrt{5} + \sqrt{x}) (\sqrt{5} - \sqrt{x})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{5 + x} (\sqrt{5} - \sqrt{x})}{{\sqrt{5}}^{2} - \sqrt{5 x} + \sqrt{x \cdot 5} - {\sqrt{x}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{5 + x} (\sqrt{5} - \sqrt{x})}{5 - x}$

Step 5

Apply the distributive property.

$\frac{\sqrt{5 + x} \sqrt{5} + \sqrt{5 + x} (- \sqrt{x})}{5 - x}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{(5 + x) \cdot 5} + \sqrt{5 + x} (- \sqrt{x})}{5 - x}$

Step 7

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{(5 + x) \cdot 5} - \sqrt{5 + x} \sqrt{x}}{5 - x}$

Step 8

Combine using the product rule for radicals.

$\frac{\sqrt{(5 + x) \cdot 5} - \sqrt{x (5 + x)}}{5 - x}$

Step 9

Reorder factors in $\sqrt{(5 + x) \cdot 5} - \sqrt{x (5 + x)}$ .

$\frac{\sqrt{5 (5 + x)} - \sqrt{x (5 + x)}}{5 - x}$