Radicals
Roots of Real Numbers (7.4)
Products & Quotients of Radicals (7.5)
Target Goals: Simplify radicals having various indices;
use a calculator to estimate roots of numbers; simplify
radical expressions using multiplication & division;
rationalize the denominator of a fraction containing a
radical expression
Radicals  not just square roots!
Radical Sign
Index
n
512
Radicand
(the number under the
radical sign)
b = radicand
n = index
n
b
b>0
b<0
b=0
n is even
2 real roots:
+&-
No real
root
0
n is odd
1 + real
roots
1 – real
root
0
SIMPLIFYING RADICAL EXPRESSIONS

A radical expression is in simplified form when the
following conditions are met:




The index is as small as possible. ***
The radicand contains no factors (other than 1) that are the
nth powers of an integer or polynomial.
The radicand contains no fractions.
No radicals appear in the denominator.
PRODUCT PROPERTY OF RADICALS
n
a b  a  b
n
a  b  a b
n
n
n
n
QUOTIENT PROPERTY OF RADICALS
n
n
n
a

b
a

b
n
n
n
a
b
a
b
Ex 1) Find the following/Simplify:
2


13x
 169x   169xxxx  13xx
4
 121a b   121aaaaaabb  11aaab
6 2
 11a 3b
Ex 2) Find the following/simplify

 8 x  3
2

8x  38x  3
  8x  3
 8x  3
3
3

2nnn

2n
8n  8nnnnnnnnn
4
m  mmmm  m
9
4
3
4
Ex 3) Find the following using your
calculator:
3
339  6.973
 670  8.750
3
3
8.1  2.008
Calculator Tips:
For cubed roots/index = 3
“MATH” button, option “4”
For indices larger than 3
* On the home screen, press the index #
* “MATH” button, option “5”
Ex 4) Simplify:
12c d  4  3  c  c  c  d  d
 2  2  3  ccccccddd
6
2
3
 2cccd 3d
 2c d 3d
3
2
2
2
Ex 5) Simplify:
3
27 y z  3 y y y y z z z
12 7
3
3
3
3
3
3 3 3
 3 3  3  3yyyyyyyyyyyyzzzzzzz
 3yyyyz
 3y z
23
4 23
z
z
Ex 6) Simplify:
9
4
a a b a 4 ab a 4 ab a 4 ab
 2
 2



3
2
5
b
b b
bb
b b bb
b
a
Ex 7) Simplify:
3
3 5 22 2 y  y  y  y
3



4 y 5 4y 5 2  2y 5 2  2  2  y  y  y  y
5
5

5
5
5
5
3 2  2  2  y  y  y  y
2 2 2 2 2 y  y  y  y  y
24 y

2y
4
Ex 8) Simplify:
6 8c d  4 2cd
3
5
3
 24 8c d  2cd
3
 24 8  2c cd d
3
3
5
3
5
 24 2  2  2  2ccccdddddddd
 24  2  2ccdddd
 96c d
2
4
Ex 9) Simplify:
2 8x y  3 2x y
4
3
2
4
5
2
 6 16x y
8
4
4
 6 4 2  2  2  2xxxxxxxxyyyy
 6  2xxy
 12x y
2