2. The Product Rule for Radicals
n n n ab a b
100 425 4 25 25 10
1000 8 125 8 125 2 5 10 3 3 3 3
3 3 4 3 3 3 3 240 2 35 2 235 2 30
3. The Quotient Rule for Radicals
n
n
n
a
b
a
b
8
11
64
121
64
121
27 3
y x
3 4
x
3 3 3
y x
3 4
x
y x
27 (4 )
x x
y
x
2
2
2 (4 )
2
3 3 3
3
2
3
3
2
3
4. A Radical Expression is Simplified When:
1. Each factor in the radicand is to a power less than the
index of the radical
2. The radicand contains no fractions or negative numbers
3. No radicals appear in the denominator of a fraction
3 2 4
16
x y
3 2
y x y
2 2
5 4
a a
5 4
2
2
a a
2
2
2
a
a
32
2
x
x
x
3 2
3
3
2 2
6. Remove all perfect nth root factors from the
radicand (variables)
9 8 8 4
m m m m m
m m
5 4 2
a a a a a
128 64 2
8 2
3 5 3 3 2 3 2
x x x x x
24 8 3
2 3
5 9 5 5 5 5 4 5 4
a b a b a
ab a
7. Leave no fractions or negative
numbers in the radicand
x x 2
2
x 2
x
3 4 3 3 3 2 3
y y y y y y y
y x
3 5 3 5 3 5
3 4
x
3 3 3
y x
3 4
x
y x
27 (4 )
x x
y
27
x
2
2
2 (4 )
2
2
4
2
2 2
3
3 3 3
3
2
3
3
2
3
8. No radicals remain in the denominator
of a fraction
5
xy
432
45
3 3 4 3 3 3
45
xy
3 5
432
3
2
2 2
54 27 2 3 2
8
8
9 3
5
5
x x x x x
x
x
x
x
y y
x
x
9. Adding & Subtracting Like Radicals
Each term must have a radical with identical
index and radicand
Law of distribution allows combining or
factoring
Like radicals:
4 4 3 2 and 7 2 2x 3x and 11 3x
Unlike radicals (cannot combine)
5 4 3 5 and 2 3 2y 3x and 4 3x