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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)

Square and Square Root Inequalities

 

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