# 1.8 – Solving with Roots and Powers

Objectives

• Apply the laws of exponents and roots to isolate variables from constants in an equation or inequality.
• Express the solution to an equation or inequality involving roots and powers as an absolute value.
• Map absolute value on the number line to determine the solution set.
• Ask the questions necessary to turn real-life problems into mathematical sentences.

Key Terms

• Absolute Value – A number’s distance from zero on a number line.
• The result of absolute value is ALWAYS positive!  Written |3| = 3 and |-3| = 3.
• If a variable is inside the absolute value bars, then the answer (result) is a positive or negative value:
• $\left| x \right| = 10$, so…
• $x=\pm10$
• Because, if you substitute +10 or -10 for x (inside the absolute value bars), you get |10| = 10 and |-10| = 10.
• There are NO SOLUTIONS to: $\left| x \right| = -10$ because the result of absolute value is ALWAYS positive!
• Simplify – To rewrite an expression as simply as possible.
• Square – To multiply a number by itself. The result is called the square of the number.
• Square Root – A factor of a number that, when squared, gives the number.
• The square root of b can be written $\sqrt{b}$ or $b^\frac{1}{2}$.
• They are the SAME thing, so: $\sqrt{b}=b^\frac{1}{2}$.
• Squared – Multiplied by itself. This is often shown with an exponent of 2.

Notes

• For any positive number or expression:
• The square of a square root is itself, because they cancel each other out.
• The result is ALWAYS POSITIVE
• $(\sqrt{x})^2=x$
• $(\sqrt{d})^2 = (d^\frac{1}{2})^2 = d$ because… $d^\frac{1}{2}\bullet^\frac{2}{1} = d^\frac{1}{1} = d^1 = d$
• ex. $(\sqrt{35})^2=35$
• ex. $(\sqrt{18.7x})^2=18.7x$

• Steps for Solving Square Root equations
• The solution to  $\sqrt{x}$ is ALWAYS POSITIVE
• Step 1:  Write the equation
• Step 2:  Square both sides (power of 2)
• Step 3:  Simplify
• Example
• Step 1:  $\sqrt{x} = 3$
• Step 2:  $(\sqrt{x})^2 = 3^2$
• Step 3:  $x = 9$
• 9 is POSITIVE because you cannot take the square root of a negative number

• For any number or expression:
• The square root of a square is the absolute value of the square, so the result is a positive and negative value:
• …because a negative times a negative is a positive
• …because a positive times a positive is a positive
• $\sqrt{\left({x}^{2}\right)} = \left| x \right| = \pm x$
• ex. $\sqrt{\left({15}^{2}\right)} = \left| 15 \right| = \pm15$
• ex. $\sqrt{\left({-5}^{2}\right)} = \left| -5 \right| = \pm5$

• Steps for Solving Squared Equations
• Step 1: Write the equation
• Step 2: Take the square root of both sides
• Step 3: Simplify
• Step 4: Write as a solution
• Example
• Step 1: $x^2 = 36$
• Step 2: $\sqrt{\left({x}^{2}\right)} = \sqrt{36}$
• Step 3: $\left| x \right| = 6$
• Step 4: $x=\pm6$
• Because:  $(-6)\bullet(-6) = 36$ and $(6)\bullet(6) = 36$

• Copy these examples and titles (below)