1. Simplifying Square and Cube Roots Objective: To simplify and estimate the value of square and cube roots.

2. Perfect Squares  The square root of a number is one of two equal factors of a number.  Perfect squares can be used to simplify square roots of rational numbers. A perfect square is found by multiplying a number with itself.  Let’s make a list of the first fifteen perfect squares.

3. Perfect Squares 11 44 99  16  25  36  49  64  81  100  121  144  169  196  225

4. Perfect Cubes  The cube root of a number is one of three equal factors of a number.  Perfect cubes can be used to simplify cube roots of rational numbers. A perfect cube is found by multiplying a number with itself twice.  Let’s make a list of the first fifteen perfect cubes.

5. Perfect Cubes 11 88  27  64  125  216  343  512  729  1000  1331  1728  2197  2744  3375

6. Example 1:  Simplify each root. .

7. Signs of Roots  If there is a negative sign (-) in front of the square or cube root, your answer will be negative.  If there is a positive or negative sign (±) in front of the square or cube root, the answer will be positive or negative.  You cannot take the square root of a negative number, but you can take the cube root of a negative number. The cube root of a negative number will always be negative.

8. Example 2:  Simplify each root. .. .

9. Estimating Roots  You can estimate roots that are not perfect squares or perfect cubes.  Find the perfect square or cube closest to the number under the root. This number may be smaller or larger than the given number, but make sure it is the closest.  Take the square or cube root of that number for your estimate.

10. Example 3:  Estimate each root to the nearest whole number. 25 36 6 8 27 2 100 121 10