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Math 1
Name ____________________________
Notes Radical Expressions: Simplify Multiply & Divide
I. Square Root of a number
If b2 = a, then b is the square root of a.
All positive real numbers have two square roots: a positive square root (or principal root) and
a negative square root. Square roots are written with a radical symbol
. The number or
expression inside a radical symbol is the radicand.
Index RadicandPower
Positive square root
9= 3
Negative square root
9 = -3
Both  9 =  3 = 3 and -3
Do negative numbers have square roots? ___________________________________________
Is there a number that has only one square root? _____________________________________
II. nth root of a number
b is the nth root of a number a if bn  a
If n is an integer greater than one, then the nth root of a is the number whose nth power is a.
There are two notations for the nth root of a.
n
a
1
an
Where n is called the index of the radical
is called the radical symbol
a is called the radicand
n
a is the radical form of the nth root of a
1
an
is the exponential form of the nth root of a
Let n be an integer greater then 1 and let a be a real number :
If n is odd, then a has one real nth root n a
If n is even and a > 0, then a has two real nth roots  n a
If n is even and a = 0, then a has one real nth root n 0
If n is even and a < 0, then a has no real nth roots
An expression containing a radical symbol is called a radical expression.
III. Simplifying a Single Radical Expression
A radical expression in simplest form:
 No perfect square factors other than 1 are in the radicand
 No fractions are in the radicand
 No radicals appear in the denominator of a fraction
1. If the number or variable is not perfect, then break it down into a perfect
square times a non-perfect.
Note: The non-perfect can’t contain a factor that is perfect
Example 1:
48 x 4 y 3  16  3  x 4  y 2  y
Perfects: 16, x 4 , y 2
Non-perfects: 3, y
2. Take the root of the perfects and leave the remaining non-perfects under the radical.
16  3  x 4  y 2  y  4 x 2 y 3 y
Note: A variable is perfect if the index will divide into the exponent with no remainder.
Examples:
2.
x4  x2
Hint: divide the power (4) by the index (2)
4  2  2 there are no
leftovers so x 2 comes out of the radical with nothing left inside the radical.
3.
x17  x8 x
4.
75 x9  25  3  x8  x  5 x 4 3x
Try these: 5.
Hint: power  index  17  2 = 8 with 1 left over so there is 1 x left in the radicand.
81x10
6.
27a3b2
7.
12ab2c12
8.
x25
IV. Adding and Subtracting Radical Expressions
Similar Radicals: Radicals are similar if they have the same index and the same radicand.
Examples: 9. 2 3 and 4 3 are similar
10. 2 3 and 5 2 are not similar
11. 2 x and 4 x are similar
12. 2 m and 5 n are not similar
We add and subtract radical expressions in the same way we add and subtract polynomials
by combining similar (like) terms.
To add or subtract radicals:
1. Simplify each radical
2. Combine like terms (add the coefficients of the radicals not the radicands)
Try these: 13. 9 x  5 x
14.
8a  2a
15.
4a3  9a3
V. Multiplying Radical Expressions
Product Property:
a b =
 a   b  when a, b > 0
The square root of a product equals the product of the square roots.
ab  a  b when a and b are positive numbers.
To multiply radicals:
1. Multiply the outside terms (coefficients) with the outside terms and multiply the
inside terms (radicands) with the inside terms.
2. Simplify the resulting radical.
Note: In cases where the radicands are large numbers or perfect squares, it might be
easier to simplify the radicals first and then multiply. But you still have to make sure the
final answer is as simple as possible.
Examples:
16. 2a  8a
17. 5 18
 16a 2  4a
 5 9  2  5  3 2  15 2
18. 3 5  7 15
 x10  x5
 21 75  21 25  3  21  5 3  105 3
These use the distributive property:
20. 2 2  8

x3  x 7
19.

 2  2  2  8  4  16  2  4  6
21.
3a

9a  16a



= 3a 3 a  4 a  3 3a 2  4 3a 2
 3a 3  4a 3  a 3
These must be foiled:
2 3 2 3
22.



 4  3 2  3 2  9  2  9  7


23. 2 5  6 3  7
6 5  2 35  18  6 7
Try these:
24. 2 z  2 z
25. 3 7  14
26.  4 6a2  2 12a3
27. 3 7 x 4 x  7

29. 4  7 4  7
28. 2 6 x  4 3x







VI. Dividing Radical Expressions
Quotient Property:
a
=
b
a
when a, b > 0
b
The square root of a quotient equals the quotient of the square roots of the numerator and
a
a
denominator.
when a and b are positive numbers.

b
b
To divide radicals:
1. See if the denominator is perfect, then simplify the outside terms (coefficients)
with the outside terms and the inside terms (radicands) with the inside terms.
2. See if the fraction can be reduced to get a perfect in the denominator.
Examples:
15
30.
16

31.
15
15

4
16
6
24
33.

6

24
34.
6

64
1 1

4 2
2 5
64
2 5
5


8
4
32.
 2
14
18

35.
27

29
16
8
7
7

9
3
8 x5
2x
 4 x4  2 x2
Try these:
35.
4x
49 x
36.
37.
3x3
4x2
38.
16 x 2
64 x 4
32
50
Remember to check and be sure your final answer is in simplest radical form
1. No perfect squares (other than 1 in radicand)
2. No fractions in radicand
3. No radicals in the denominator of a fraction