Section 7.1: Extracting and Simplifying Roots Worksheet 7.1
We'll get to the serious business of square roots.
But first, some bad mathematical humor:
$$\sqrt{-1}\,\,\,\,2^3\,\,\,\, \Sigma \,\,\,\, \pi \mbox{ ... and it was DELICIOUS!}$$
Solve the equation $w^2=4$.
Solve the equation $z^2=75$.
A Property of Square Roots
$$\sqrt{a\cdot b}=\sqrt{a} \cdot \sqrt{b}$$
"A square root of a product is a product of the square roots."
Example: Simplify $\sqrt{75}$
Example: Simplify $\sqrt{48}$
Example: Simplify $\sqrt{1980}$
Awesome! We can now solve basic quadratic equations.
$\xi^2=18$
$5\phi^2=2750$
$3a^2 - 150 = 0$
Another Property of Square Roots
$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$$
"A square root of a quotient is a quotient of the square roots."
Example: $\sqrt{\frac{81}{49}}$
Example: $\sqrt{\frac{8}{25}}$
Example: $\frac{\sqrt{2475}}{\sqrt{11}}$
Example: $\frac{\sqrt{17}}{\sqrt{153}}$
Rationalizing Denominators: Sometimes we don't want square roots in the denominator.
Question: Why?
Answer: Ask Dr. Math!
Example: Rationalize the denominator. $\frac{21}{\sqrt{3}}$
Example: Rationalize the denominator. $\frac{\sqrt{17}}{\sqrt{5}}$
Example: Rationalize the denominator. $\frac{\sqrt{7}}{\sqrt{2}}$
And what is our reward for going to all this trouble to learn how to deal with square roots? ...
Answer: We get to solve more equations! :D
Example: $7\gamma^2-54=0$
Example: $(4y - 5)^2-50=0$