Question

what is the reduced radical form of each expression?

1. \\(\frac{3 - \sqrt{7}}{3 - \sqrt{5}}\\)
2. \\(\frac{-5x}{3 - \sqrt{x}}\\)

Explanation:

Step1: Rationalize denominator for Q6

Multiply numerator and denominator by $3+\sqrt{5}$
$\frac{3-\sqrt{7}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{(3-\sqrt{7})(3+\sqrt{5})}{(3)^2-(\sqrt{5})^2}$

Step2: Simplify denominator for Q6

Calculate denominator using difference of squares
$\frac{(3-\sqrt{7})(3+\sqrt{5})}{9-5} = \frac{(3-\sqrt{7})(3+\sqrt{5})}{4}$

Step3: Expand numerator for Q6

Distribute terms in numerator
$\frac{9 + 3\sqrt{5} - 3\sqrt{7} - \sqrt{35}}{4}$

Step4: Rationalize denominator for Q7

Multiply numerator and denominator by $3+\sqrt{x}$
$\frac{-5x}{3-\sqrt{x}} \times \frac{3+\sqrt{x}}{3+\sqrt{x}} = \frac{-5x(3+\sqrt{x})}{(3)^2-(\sqrt{x})^2}$

Step5: Simplify denominator for Q7

Calculate denominator using difference of squares
$\frac{-5x(3+\sqrt{x})}{9-x}$

Step6: Distribute numerator for Q7

Expand the numerator terms
$\frac{-15x -5x\sqrt{x}}{9-x}$ or $\frac{15x +5x\sqrt{x}}{x-9}$

Answer:

1. $\frac{9 + 3\sqrt{5} - 3\sqrt{7} - \sqrt{35}}{4}$
2. $\frac{-15x -5x\sqrt{x}}{9-x}$ (or equivalent $\frac{15x +5x\sqrt{x}}{x-9}$)