This document defines radicals and rational exponents. It defines the principal square root and nth root of a real number. It provides rules for simplifying radicals using the product, quotient, and power rules. It also defines rational exponents where the denominator represents the root and the numerator represents the exponent. Examples are provided to demonstrate simplifying expressions using these rules and definitions.

1. Radicals and
Rational Exponents

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 = Öa, is the principal square
root of a.

3. Square Roots of Perfect Squares
a2 = a

4. The Product Rule for Square Roots
• If a and b represent nonnegative real
number, then
ab = a b and a b = ab
• The square root of a product is the product
of the square roots.

5. Text Example
• Simplify a. Ö500 b. Ö6xÖ3x
Solution:
b. 6x × 3x = 6x × 3x
= 18x2 = 9x2 ×2
= 9x2 2 = 9 x2 2
= 3x 2
a. 500 = 100 ×5
= 100 5
= 10 5

6. The Quotient Rule for Square Roots
• If a and b represent nonnegative real
numbers and b does not equal 0, then
a
b
= a
b
and
a
b
= a
b
.
• The square root of the quotient is the
quotient of the square roots.

7. Text Example
• Simplify:
Solution:
100
9
= 100
9
= 10
3
100
9

8. Example
• Perform the indicated operation:
4Ö3 + Ö3 - 2Ö3.
4 3 + 3 - 2 3 = 3 3
Solution:

9. Example
• Perform the indicated operation:
Ö24 + 2Ö6.
Solution:
+ =
24 2 6
+ =
2 6 2 6 4 6

10. Definition of the Principal nth Root
of a Real Number
n a = b means that bn = a
• If n, the index, is even, then a is
nonnegative (a > 0) and b is also
nonnegative (b > 0) . If n is odd, a and b
can be any real numbers.

11. Finding the nth Roots of Perfect
nth Powers
If n is odd, n an = a
If n is even n an = a.

12. The Product and Quotient Rules
for nth Roots
• For all real numbers, where the indicated
roots represent real numbers,
n a n b = n ab and
n a
n b = a
b
n , b ¹ 0

13. Definition of Rational Exponents
a1 / n = n a.
Furthermore,
a-1/ n = 1
a1/ n = 1
n a
, a ¹ 0

14. Example
• Simplify 4 1/2
1
= =
42 4 2
Solution:

15. Definition of Rational Exponents
am/ n = (n a )m = an m .
• The exponent m/n consists of two parts: the
denominator n is the root and the numerator
m is the exponent. Furthermore,
a-m/ n = 1
am/ n .

16. Radicals and
Rational Exponents