Download Simplifying Radicals

Multiplying & Dividing
Radicals
Operations with Radicals
(Square Roots) Essential
Question
How do I multiply and divide
radicals?
The multiplication property is
often written:
a b
ab
or
ab  a b
*
To multiply radicals: multiply the
coefficients (the numbers on the outside)
and then multiply the radicands (the
numbers on the inside) and then simplify
the remaining radicals.
Multiply and then simplify
5 * 35  175  25 * 7  5 7
2 8 * 3 7  6 56  6 4 *14 
6 * 2 14  12 14
2 5 * 4 20  20 100  20 *10  200
 5
2

5* 5 
25  5

7* 7 
49  7

8* 8 
64  8

x* x 
x 
 7
2
 8
2
 x
2
2
x
Trick Question
Is there a number in front of
√7 ?
Hint: Is there a number in front of
x?
The answer is 1. Just as there is a “ghost 1”
in front of the x, there is also a “ghost 1” in
front of the √7
A related property says that
square roots are also
distributive over division:
a

b
a
b
or
a

b
a
b
To divide radicals: divide
the coefficients (the
numbers on the outside),
divide the radicands (the
numbers on the inside) if
possible, and rationalize
the denominator so that
no radical remains in the
denominator
56

7
8
4*2  2 2
This cannot be
divided which leaves
the radical in the
denominator. We do
not leave radicals in
the denominator. So
we need to
rationalize by
multiplying the
fraction by something
so we can eliminate
the radical in the
denominator.
6

7
6
*
7
42

49
7

7
42
7
42 cannot be
simplified, so we are
finished.
This can be divided which
leaves the radical in the
denominator. We do not
leave radicals in the
denominator. So we need
to rationalize by
multiplying the fraction by
something so we can
eliminate the radical in the
denominator.
5

10
1
*
2
2
2
2

2
This cannot be divided
which leaves the radical in
the denominator. We do
not leave radicals in the
denominator. So we need
to rationalize by multiplying
the fraction by something
so we can eliminate the
radical in the denominator.
Reduce
the
fraction.
3

12
3
*
12
3

3
3 3

36
3 3

6
3
2