Honors Advanced Algebra
7.4 & 7.5 – nth Roots and Operations with Radical Expressions
Notes
Warm-Up:
List the perfect squares from 1 to 225:
Name: ________________________
Date: _________________________
Period: ________
List the perfect cubes from 1 to 1000:
Target Goals: Simplify radicals.
Simplify radical expressions.
Use a calculator to approximate radicals.
Multiply radical expressions.
nth ROOTS
For any real numbers a and b, and any positive integer n: if if a
2  b a n  b
, we say “a is the _________________________ of b,” and write ______________; and
, we say “a is the _________________________ of b,” and write ______________.
The symbol n
indicates an n th root. radical sign index 4
81 radicand
Some numbers have more than one real root. When there is more than one real root (index is even), the nonnegative root is called the principal root.
Can we take the square root of a negative number?
Can we take the cube root of a negative number?
Simplify.
Ex 1. 81 Ex 2.

144 Ex 3.

4

8 Ex 5.

9 Ex 4.
3
SIMPLIFYING RADICAL EXPRESSIONS
We can use the Product Property of Radicals to simplify radicals whose radicands are nth powers.
Product Property of Radicals: For any real numbers a and b and any integer n  1 , n ab  n a  n b . If n is even, both a and b must be nonnegative.
Simplify.
Ex 6. 25 a b Ex 7. 3
 3 6
64 m p Ex 8. 4 16
 x

1

8

USE OF ABSOLUTE VALUE
Recall that the principal root (even index) is nonnegative, therefore account for the possibility that x might be negative. In general: n a n  a when n is even.
** We will ignore the use of absolute value in simplifying radicals when the variables are given as nonnegative.
Simplify - variables not given as nonnegative!
Ex 9.
4
Ex 10.
4 6 8
36 x y z
We can also use the Product Property of Radicals to simplify radicals whose radicands are not n th powers.
Simplify. Assume each variable is nonnegative.
Ex 11. 20 ab c Ex 12. 45 k m Ex 13. 3 32
4 5 6 x y z
Another use of the Product Property of Radicals is multiplying radicals with like indices.
Simplify. Assume each variable is nonnegative.
Ex 14. 6 cd

10 d Ex 15.
3
9 rst
2  3
3
Roots of real numbers that retain a radical sign in simplest form are irrational. Calculators can be used to approximate their value.
Use a calculator to approximate each value to three decimal places.
Ex 16. 7
You try it …
Ex 17.
3 100
Ex 18.
5 30 a.

121 a b
2 b.

196 x
4
 y

3

2 c.
3 
125 m n
6 d. 48 c d
10 e. 3 gh

21 g f.
Assignment #1: pg 434 #13-29 odd, 39-43 odd, 47-51 odd; pg 443 #18-21, 26-29
3 4 xy
2 
3 4 xy
2