Multiplication and Division Properties of Radicals
If
and
are real numbers and b ≠ 0. Then
1)
Multiplication Property of Radicals:
=
2)
Division Property of Radicals:
=
•
Simplified Form of a Radical
A radical is simplified if all of the following conditions are met:
1) All factors of the radicand have powers that are less than the index.
2) The radicand has no fractions.
3) There can be no radicals in the denominator.
4) One is the only common factor of the index & exponents in the radicand.
If the power of a factor in the radical is larger than the index, divide that
power by the index. The quotient gives you how many factors come out
from under the radical and the remainder gives you how many factors are
left under the radical.
Simplify. Assume the variables represent positive real numbers:
72
Ex. A
Ex. D
€ A)
B)
D)
Ex. E
3
80
Ex. C
Ex. F
72 = 62 • 2 = 62 • 2 = 6 2
Since 80 = 16•5 = 24•5, then
4
4
80 = 24 •5 = 24 • 4 5 = 2 4 5
€Since 7 ÷ 2
€ = 3 R€1 and 3€÷ 2 = 1 R 1, then
€
€
4
Since 72 = 36•2 = 62•€2, then
4
€ C)
Ex. B
=
€
=
= 5x3y
•
€
€
Since 4 ÷ 3 = 1 R 1 and 8 ÷ 3 = 2 R 2 and 54 = 27•2 = 33•2, then
=
=
•
= 3ab2
E)
Since 4 ÷ 4 = 1, 9 ÷ 4 = 2 R 1, 12 ÷ 4 = 3, and 32 = 16•2 = 24•2,
then 3
=3
=3
•
2
= = 3•2xy z3
2
= 6xy z3
.
F)
Since 9 ÷ 3 = 3, 6 ÷ 3 = 2, and 48 = 16•3 = 24•3 = 23•2•3, then
=
=
•
= 2x3b2
Simplify. Assume the variables represent positive real numbers:
Ex. G
Ex. H
Ex. I
Ex. J
Ex. K
G)
=
H)
=
I)
=
=
=
=
=
.
=
=
=
.
J)
K)
= r2
=
=
=
=
=
The indices are not the same so we cannot use our properties. But,
=
= x1/3 + 1/4 – 1/5 = x20/60 + 15/60 – 12/60 = x23/60 =
Simplify. Assume the variables represent positive real numbers:
1)
€
75
2)
€
500
Simplify. Assume the variables represent positive real numbers:
3)
5
− 486
4)
3
6)
–4
r 6s11t19
€
€
5)
7)
€
5
5
7
8)
700000
€
9)
10)
5
x•
7
4
x
x