December 12, 2013
5-5 Roots of Real
Numbers
AA
Square Root:
* For any real numbers c and d, if c2 = d,
then c is a square root of d.
* Square roots and squaring are inverse
operations.
* √
72 = 49, 7 is a square root of 49
(-7)2 = 49, -7 is a square root of 49
nth Root:
* For any real numbers c and d, any positive
integer n, if cn = d, then c is an nth root of
d.
* Raising a number to the nth power is the
inverse of the nth root of a number.
* Symbol n√
24 = 16, 2 is a fourth root of 16
December 12, 2013
Principal Root:
* The nonnegative root of a number.
n
* √ b indicates the principal nth root of b.
If n is odd and b is negative, there will not
be a nonnegative root. In this case the
principal root is negative.
√ 25 = 5
principal square root of 25
- √ 25 = -5 opposite principal square
root of 25
± √ 25 = ±5 both square roots of 25
3
√ -125 = -5 principal cube root of -125
4
- √ 81 = -3 opposite principal fourth
root of 81
December 12, 2013
n
th
Real n roots of b, √ b
n
or - √ b .
When n is even:
If b > 0, then one positive and one
negative root.
If b < 0, then there are not real roots.
If b = 0, one real root, and that is 0.
When n is odd:
If b > 0, then one positive root and no
negative root.
If b < 0, then there are no positive roots, one
negative root.
If b = 0, one real root, and that is 0.
December 12, 2013
Simplify Using Absolute Value:
* When you have even roots you will need
to take the absolute value of x to identify
the principal root.
8
√
4
√
x8
81(a + 1)
12
=|x|
=
4
√ 3(a + 1)3 4
= 3 | (a + 1)3|
Calculators can give you approximations
for irrational numbers.
5
(n) ^ (1/5) = √ n
4
√ 216
=
December 12, 2013
1. Simplify.
√16x6
a. ±
c.
b. -
√(q3 + 5)4
5
√ 243a10b15
d. √
-4
2. Simplify
6
a. √
t6
b.
5
√ 243(x + 12)15
December 12, 2013
5 - 5 Roots of Real Numbers
page 247 -249
# 16 - 23, 28 - 34, 44 - 48, 60, 61,
67 - 70, 72, 74, 78, 80 (30 problems)