1. P. 3 Radicals and Rational Exponents Q: What is a radical
P.3 Radicals and Rational Exponents Q: What is a radical? What is a rational number? A: A “radical” involves a root symbol, whereas a “rational number” involves a fraction.

2. Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative number b such that b2 = a, denoted by b = a, is the principal square root of a.
That is: 4=2 (since 2 squared = 4), not –2
(even though (-2) squared also = 4).

3. Square Roots of Perfect Squares
Ex: Simplify (-3)2
Ex: Simplify x2
Ex: Simplify -32
Ans: 3, |x|, and “not a real number” or 3i

4. The Product Rule for Square Roots
If a and b represent nonnegative real number, then
The square root of a product is the product of the square roots.
Ex: Compare and draw conclusions:
9+16 vs. 9*16

5. Ex: Simplify
a. 500
b. 6x3x
c. 108x6y11
Ans: a) b) c)

6. The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers and b does not equal 0, then
The square root of the quotient is the quotient of the square roots.
Ex: Simplify (ans: 10/3)

7. Example Ex: Perform the indicated operation: Ans:
We can only add radical expressions if they contain “like terms”:
The same number must be under the radical sign (the radicand), and it must have the same index. Then just like ordinary “like terms” we add the COEFFICIENTS and KEEP THE “LIKE” parts the SAME.
Ex: Perform the indicated operation:
Ans:

8. Ex: Perform the indicated operation:
724 + 26 =
Ans:

9. Definition of the Principal nth Root of a Real Number
If n, the index is:
even, and a is nonnegative (a > 0) then b is also nonnegative (b > 0)
Ex:
odd, a and b can be any real numbers with the same sign (+ or -)
Q: What would we write if n is even and a is negative?
(Ans: “not a real number”.)

10. Finding the nth Roots of Perfect nth Powers
It is only “necessary” to use the absolute value symbol if you are finding the even root of a variable (unknown).

11. Ex: Simplify each of the following:
³(-2)3
(-2)2
3-8x7y11
416x8y3
x10 32
Ans: -2 2

12. The Product and Quotient Rules for nth Roots
For all real numbers, where the indicated roots represent real numbers,
Q: Do you remember for what operation(s) you may NOT separate ( or reverse to put together) the numbers? (A: sum or difference.)

13. Definition of Rational Exponents
The denominator of the rational exponent becomes the
INDEX of the radical expression.

14. 4 ½ (-8)(2/3) (250x9y7)1/3 (125x6)2/3 Ex: Simplify the following: Ans:

15. Definition of Rational Exponents
The exponent m/n consists of two parts: the denominator n is the root and the numerator m is the exponent. Furthermore,

16. Example: Simplify 2(-8x12)-(2/3)
Ans:

17. Rationalizing the Denominator
ONE TERM in the denominator: simplify, then multiply by whatever is needed to make a perfect root (ONE TERM).
Ex:
TWO TERMS in the denominator (one is a square root): simplify, then multiply by the conjugate (TWO TERMS).

18. Simplified form for Radical Expressions:
NO radical sign in the denominator
NO fractions under the radical sign
NO exponents greater than the index under the radical sign
The index is reduced as low as possible