𝑁 𝑇𝐻 Roots and Real Exponents
0.4
Obj: simplify expressions in radical form and simplify expressions in exponential form.
DOL: I will correctly solve 80% of problems
Do Now: Simplify
2) √𝑥 4
1) √50
n
a
3)
n = index
1
4)
√3
3 5
a = radicand
Number and Types of Roots
Odd index: 1 real root
Even index, positive radicand: 2 real roots (± answer)
Even index, negative radicand: 0 real roots
Radicand of 0: 1 real root
Find the roots
1)
4
81
2)
5) 6 1
3
6) 3 125
Of nth roots:
Product Property:
Quotient Property:
n
ab  n a  n b
n
a

b
n
a
n
b
 125
3)
6
 729
4)  4 256
*When you find the even root of an even power and the result is an odd power, you must use the absolute value
of the result to make sure that the answer is nonnegative.*
Simplify
6
4
1) √𝑛18
4)
2) √81(𝑎 + 1)12
5
√−𝑝10 𝑞 7
Definition of b1/n : b1/n =
5)
n
3
3) √63𝑦 3
x3
7
6)
4
x8
3
b ; n > 1, except where b < 0 and n is even
Definition of Rational Exponents: For any nonzero real number b and any integer m and n with n > 1 except
when b < 0 and n is even
b
m
n
 n bm 
 b
m
n
𝐩𝐨𝐰𝐞𝐫
*
𝐫𝐨𝐨𝐭
Write in radical form and simplify
10
1)
√32
6
√2
3
3)
164
1
164
4
2) √𝑥 21 𝑦15
4
√25
4) 6 .
√5
∗
Practice:
Evaluate each expression.
a) −√81
4
b) √
16
6
c) √−243
625
Simplify each expression
8
a) √𝑏 24
6
e) √𝑥 31 𝑦 20
6
b) √64(𝑦 + 2)30
1
1
1
3
c) √48𝑥 5
2
f) (2𝑥 4 𝑦 3 ) (3𝑥 4 𝑦 3 )
d) √
4
√27
g) 3
√81
5
𝑥2