Mastery Alg 1
Reteach 7-5
Name: _________________________
Reteaching
7-5
Rational Exponents and Radicals
When you understand the equivalence of radicals and rational exponents, you can represent a
radical expression using rational exponents.
If the nth root of a is a real number and m and n are positive integers, then
1
m
n
a n  n a and a n  a
m
You can also represent an expression with rational exponents as a radical expression.
Sometimes converting from one form to the other makes it easier to simplify an expression
involving roots.
Problem
3
What is 8 2 in radical form? Simplify, if possible.
2
3
Rewrite the expression in radical form.
3
Simplify.
8 3  82
 82  3 64  3 4  4  4  4
Write each expression in radical form, then simplify if possible.
1. 36
1
5
2
2
2
3
2. 4
3.
(27 x)
5
4. 18b
3
Problem
What is
4
(16 x)3 in exponential form? Simplify, if possible.
3
4
3
(16 x )  16 x
3
4
4
3
4
 16 x  8 x
Rewrite the expression in radical form.
3
4
Simplify.
Write each expression in exponential form, then simplify if possible.
5.
3
82
6.
3
8x
7.
3
64x 7
8.
3
(27 x)2
You can use the properties of exponents to simplify expressions with rational
exponents. When simplifying radical expressions, you can combine like terms.
Problem
 1  1
What is the simplified form of the expression,  a 4   a 4  ? Use the properties of
  
exponents, and then write the expression in radical form.
1
1
1
 14   14 
4
4
2
a
a

a

a
  
  

When multiplying like bases, add the exponents.
a
Rewrite in radical form.
Simplify each expression using the properties of exponents, and then write the
expression in radical form.
 23   1 
9.  a   a 4 
  
 12 
10. ( ab)  b 
 
1
3
 15  2
11.  5x  ( x )
 
1
1
3
12. (8 x ) (64 x) 2
Problem
What is the exponential form of the expression,
3
5
a 3 a  a  3a 2
3
5
a3  3 a5 ?
2
Rewrite the expression in exponential form. The expression
cannot be simplified further because the terms are not
like terms.
Write each expression in exponential form. Simplify when possible.
 16x  27 x 
13.
3
a 2 2 a 2
14.
15.
3
(27 x)2  4 256 x 2
16. 4 3 (3x)  (2 x)
3
3
4
6
5
4