Question

simplify the following equations. write all answers using exact values.

1. $x = \frac{2\pm6\sqrt{3}}{4}$
2. $x = \frac{3\pm27\sqrt{3}}{9}$
3. $x = \frac{-15\pm10\sqrt{3}}{5}$
4. $x = \frac{4\pm3\sqrt{3}}{2}$
5. $\sqrt{150}$
6. $\sqrt{108}$
7. $\sqrt{96}$
8. $2\sqrt{6} + 3\sqrt{54}$
9. $- 3\sqrt{7} + 4\sqrt{7}$
10. $- 4\sqrt{8} * \sqrt{10}$

Explanation:

Problem 9:

Step1: Factor out 2 from numerator

$x = \frac{2(1 + 3\sqrt{3})}{4}$

Step2: Simplify the fraction

$x = \frac{1 + 3\sqrt{3}}{2}$

Step1: Factor out 3 from numerator

$x = \frac{3(1 + 9\sqrt{3})}{9}$

Step2: Simplify the fraction

$x = \frac{1 + 9\sqrt{3}}{3}$

Step1: Factor out 5 from numerator

$x = \frac{5(-3 + 2\sqrt{3})}{5}$

Step2: Simplify the fraction

$x = -3 + 2\sqrt{3}$

Answer:

$x = \frac{1 + 3\sqrt{3}}{2}$ (and $x = \frac{1 - 3\sqrt{3}}{2}$ for the minus case, but simplified form is $\frac{1 \pm 3\sqrt{3}}{2}$)

Problem 10: