Review of Radicals MATH 017 Intermediate Algebra S. Rook

2 Overview Section 7.1 in the textbook –Find square roots –Approximate roots –Find cube roots –Find n th roots

Find Square Roots

4 Should be a review for numbers means “what number multiplied by itself gives you a”? What about the square root of a negative number? –Suppose we want to evaluate What number multiplied by itself gives you -4? There is none because the product of two numbers with the same sign is always positive! Therefore, the square root of a negative number does NOT exist in the real number system

5 Find Square Roots (Continued) Slightly different for variables –Consider

6 Find Square Roots (Continued) Thus: if a is divisible by 2 Perfect squares –Should have the first ten perfect squares memorized xx2x2 xx2x2 xx2x2 xX2X

7 Find Square Roots (Example) Ex 1: Evaluate

Approximate Roots

9 Most square roots will not evaluate to integers, but to irrational numbers Can approximate by “squeezing” the root between two perfect squares

10 Approximate Roots (Example) Ex 2: Approximate and then evaluate it using a calculator

Find Cube Roots

12 Find Cube Roots Should be a review for numbers means “what number multiplied by itself three times gives you a”? What about the cube root of a negative number? –Suppose we wish to evaluate What number multiplied by itself three times gives you -8? -2 Therefore, the cube root of a negative number exists in the real number system because the product of three negatives is negative

13 Find Cube Roots (Continued) Slightly different for variables –Consider

14 Find Cube Roots (Continued) Thus: if a is divisible by 3 Perfect cubes –Should have the first five perfect cubes memorized xx3x3 xx3x

15 Find Cube Roots (Example) Ex 3: Evaluate

Find n th Roots

17 Find n th Roots Should be a review for numbers means “what number multiplied by itself n times gives you a”? What about the n th root of a negative number? –Already saw that the cube root of a negative number exists in the real number system exists while the square root of a negative number does not –Can extend this to the general case The product of an even number of negatives is positive –Therefore, the even root of a negative number does NOT exist in the real number system The product of an odd number of negatives is negative –Therefore, the odd root of a negative number DOES exist in the real number system

18 Find n th Roots (Continued) Slightly different for variables –Consider

19 Find n th Roots (Continued) Thus: if a is divisible by n

20 Find n th Roots (Example) Ex 4: Evaluate

21 Summary After studying these slides, you should know how to do the following: –Evaluate square, cube, and n th roots –Approximate a root