MTH401 UNIT 0

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