Intermediate Algebra (11th edition)

NOTE From Example 5,we see that if a variable is raised to a power with an expo-

nent divisible by 2, it is a perfect square. If it is raised to a power with an exponent

divisible by 3, it is a perfect cube. In general, if it is raised to a power with an

exponent divisible by n, it is a perfect nth power.

SECTION 8.3 Simplifying Radical Expressions 447

OBJECTIVE 4 Simplify products and quotients of radicals with different

indexes.We multiply and divide radicals with different indexes by using rational

exponents.

FIGURE 6

NOW TRY EXERCISE 6 Simplify. Assume that all variables represent positive real numbers. (a) 2672 (b) 26 y^4

NOW TRY ANSWERS

(a) (b)

(a) 261944

237 23 y^2

NOW TRY EXERCISE 7

Simplify. 233 # 26

The conditions for a simplified radical given earlier state that an exponent in the

radicand and the index of the radical should have greatest common factor 1.

If mis an integer, nand kare natural numbers, and all indicated roots exist, then

.

kn

2 akm

n

2 am

2 knakm

Simplifying Radicals by Using Smaller Indexes

Simplify. Assume that all variables represent positive real numbers.

(a)

We write this radical by using rational exponents and then write the exponent in

lowest terms. We then express the answer as a radical.

,or

(b) Recall the assumption that

NOW TRY

24 p^2 = 1 p^22 1/4= p2/4= p1/2 = 2 p 1 p 7 0. 2

2956 = 1562 1/9 = 5 6/9= 5 2/3= 2352 2325

2956

EXAMPLE 6

CAUTION The computation in FIGURE 6is not proofthat the two expressions are

equal. The algebra in Example 7,however, is valid proof of their equality.

Results such as the one in Example 7can be supported with a calculator, as

shown in FIGURE 6. Notice that the calculator gives the same approximation for the

initial product and the final radical that we obtained.

Multiplying Radicals with Different Indexes

Simplify

Because the different indexes, 2 and 3, have a least common multiple of 6 , use

rational exponents to write each radical as a sixthroot.

Now we can multiply.

Substitute;

= 261372 Product rule NOW TRY

27 # 232 = 26343 # 264 27 = 26343 , 232 = 264

232 = 2 1/3= 2 2/6 = 2622 = 264

27 = 7 1/2^ = 7 3/6 = 2673 = 26343

27 # 232.

EXAMPLE 7

These examples suggest the following rule.