# Roots and Rational Exponents

1.  $\rhd$ Simplifying square roots (3:08) $5 \sqrt{117} = 5 \sqrt{3^2 \cdot 13} = 15 \sqrt{13}$ and $3 \sqrt{26}$ cannot be simplified.
2.  $\rhd$  Simplifying square roots of fractions (4:40) $\sqrt{\frac{1}{200}} = \frac{1}{10\sqrt{2}} = \frac{\sqrt{2}}{20}$
3.  $\star$ Simplifying square roots $\sqrt{\frac{1}{44}}$
4. $\rhd$ Simplifying sums of radicals (4:41) He uses the correct $|x| = \sqrt{x^2}$ rather than the (false) $\sqrt{x^2} = x$. Hooray!
Solve:   $\sqrt{2x^2} + 4\sqrt{8} + 3\sqrt{2x^2} + \sqrt{8} = (4|x| + 10)\sqrt{2}$
5.  $\rhd$ Simplifying differences of radicals (5:45) $4 \sqrt[4]{81x^5} - 2\sqrt[4]{81x^5} - \sqrt{x^3} = 6|x| \sqrt[4]{x} - |x|\sqrt{x}$  In this video he correctly remarks that “If $x>0$, there’s no need for the $|x|$.”
6.  $\star$ Simplifying square root expressions (with no variables) $\sqrt{\frac{6^2}{95}}$
7. $\rhd$ Introduction to rationalizing the denominator (10:17)  [Example 1] $\frac{1}{2}$
[Example 2] $\frac{7}{15}$
[Example 3] $\frac{12}{2 - \sqrt{5}}$  Sal shows why multiplying by $\frac{\sqrt{5}}{\sqrt{5}}$ doesn’t help and why multiplying by $\frac{2+\sqrt{5}}{2+\sqrt{5}}$ does — difference of squares!
[Example 4] $\frac{5y}{2\sqrt{y} - 5} = \frac{10}{4y-25}$