V3. Multiplying Radical Expressions
Recall that the root of the products is the product of the roots, or,
√ √
√
n
xy = n x n y.
And, the root of the quotient is the quotient of the roots, or,
√
r
n
x
x
n
= √
.
n y
y
In the problems that follow, the radicals cannot be simplified; however, when
multiplied, the product allows for the radical to be simplified.
√ √
Example. Simplify 5 · 5.
√ √
√
√ √
√
Using our property above, 5· 5 = 25. Since 25 = 52 , then 5· 5 = 25 =
5.
√ √
Example. Simplify 5 · 10.
First notice
√ that neither square root can be simplified, so multiply the radials
to find 50. To simplify this look for perfect squares that are factors of 50. In
this case 25 is the perfect square of 52 . Then,
√ √
√
5 · 10 = 50
√
= 25 · 2
√
= 52 · 2
√
=5 2
√
2
Example. Simplify √ .
18
Using the property above,
q √
2
1 2
then √18
=
= 13 .
3
√
√2
18
=
q
2
18
=
q
2·1
2·9
=
q
2·1
2·9
=
q
1
9.
Since
1
9
= ( 13 )2 ,
Not all problems are so clean, but the concept extendeds to larger problems.
p
p
Example. Simplfiy 3 30x2 y 5 · 3 1800x2 y.
The radicals cannot be simplified; however, when multiplied, the resulting product allows for the radical to be simplified.
1
p
3
30x2 y 5 ·
p
3
1800x2 y =
p
3
30x2 y 5 · 1800x2 y
=
p
3
54000x4 y 6
Look for cubes.
p
3
33 · 103 · 2x3 x(y 2 )3
√
3
= 3 · 10 · xy 2 2x
√
3
= 30xy 2 2x
=
p
324x7 y 3 z
Example. Simplify p
.
50x4 y 5 z 3
p
324x7 y 3 z
p
=
50x4 y 5 z 3
s
324x7 y 3 z
50x4 y 5 z 3
s
2 · 162x4 x3 y 3 z
2 · 25x4 y 3 y 2 z 2 z
Factor to cancel common factors.
4 3 3
2
x
x y z
· 162
4
3
2
xy y 2 z 2 z
· 25
Look for perfect squares.
=
s
=
s
=
162x3
25y 2 z 2
s
2 · 81x2 x
25y 2 z 2
√
9x 2x
=
5yz
Or,
9x √
=
2x
5yz
=
2
0.1
1.
4.
7.
10.
Practice Problems
√
√
5·
√
5x3 ·
p
√
3
5
√
2.
p
24a5 b2 ·
54x2 y
√
3
9a7 b8
8.
16.
19.
0.2
√
5
3.
11.
3x9
243x
p
48xy 3
√
75x
√
3
108a4 b10
√
3
500a7 b4
√
14.
17.
20.
√
18
√
2
√
1000x5
√
10x
p
175x3 y 4 z 2
p
28x5 yz
√
3
−160a11 b2
√
3
135a6 b5
15.
18.
1. 5
√
2. 10 3
√
3. 2 3 7
√
4. 10x2 2
√
5. 24x 11
√
6. 80x2 y y
√
7. 6x2 y 2 3x
√
8. 36x4 y 3 10y
√
9. 20ab2 3 2a
11. 3
12.
5
4
15.
3
2x
18.
√
9x 2x
5yz
13.
x4
9
14. 10x2
16.
√
4y y
5
17.
19.
3b2
5a
√
5y yz
2x
√
3
2a 4a2
20. −
3b
3
9.
12.
Soltuions
√
10. 6a4 b3 3 b
√
3
2·
√
3
28
p
p
6. 5 32xy 2 · 8x3 y
p
p
405x5 y 3 · 32x3 y 4
√
13.
60 ·
√
√
5. 4 3x · 2 33x
40x
2x3 y 3 ·
√
√
3
100a2 b5 ·
√
3
√
50
√
32
√
54x
√
24x3
p
324x7 y 3 z
p
50x4 y 5 z 3
160a2 b