College Algebra 9th txtbk

SECTION R–2 Exponents and Radicals 17
MATCHED PROBLEM 6
Simplify and express answers using positive exponents only. All letters represent positive
real numbers.
(A) (8) 5 3
(B) 2 5 32 4 (C) (52 4 y 3 )(21 3 y) (D)
a 8x1 2
x 2 3 b 13
Z Simplifying Radicals
The exponent properties considered earlier lead to the following properties of radicals.
Z THEOREM 4 Properties of Radicals
For n a natural number greater than 1, and x and y positive real numbers:
1. 2 n x n x
2. 2 n xy 2 n x2 n y
n x
3.
By 2n x
2 n y
2 3 x 3 x
2 5 xy 2 5 x2 5 y
4 x
A y 14 x
1 4 y
An algebraic expression that contains radicals is said to be in simplified form if all four
of the conditions listed in the following definition are satisfied.
Z DEFINITION 5 Simplified (Radical) Form
1. No radicand (the expression within the radical sign) contains a factor to a power
greater than or equal to the index of the radical.
For example, 2x 5 violates this condition.
2. No power of the radicand and the index of the radical have a common factor
other than 1.
For example, 2 6 x 4 violates this condition.
3. No radical appears in a denominator.
For example, y/ 1x violates this condition.
4. No fraction appears within a radical.
For example,
2 3 5
violates this condition.
EXAMPLE 7 Finding Simplified Form
Write in simplified radical form.
(A) 212x 5 y 2 (B) 2 6 16x 4 y 2 (C)
6
3 8x 4
(D)
12x B y