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2. Simplifying Radicals:

A radical is an indicated root of a number.
Index
3
3
3 indicates cube root, therefore 8 = 2, because 2 = 8
n p
4
4 indicates fourth root, therefore 16 = 2, because 2 = 16
4

Radical sign Radicand
Principal Root:
Example: (a) Even index
 9 has two square roots, 3 and – 3
3 is the principal root and is denoted by 9 .
 16 has two real fourth roots, 2 and – 2

2 is the principal fourth root and is denoted by 16
4
(b) Odd index
 8 has one real cube root, 2

3 8 = 2
 – 8 has one real cube root, – 2
3  8 = – 2

The simplified form of a radical
1) No prime factor of a radicand has an exponent
equal to or greater than the index.
2) No radicand contains a fraction.

3) No denominator contains a radical.

Examples:
2
3
1) (a) 250 = 5  2 = 5   2 = 25 10 = 5 10
5
3
3
3
3
(b) 3 24 = 2  3 = 2 3 3 3 = 2 3

25 25 5
2) (a) = =
36 36 6
8 3 8 2
(b) 3 = =
27 3 27 3

5 5 2 5 2
3) (a)   =
2 2 2 2

2 2 5  1 2  5   1 5  1
(b)   = =

5  1 5  1 5  1 5 1 2

Calculus 7 Unit (1)