Basic Mathematics for College Students

The rule for multiplying two fractions can be extended to find the product of

three or more fractions.

224 Chapter 3 Fractions and Mixed Numbers

Success Tip If you recognized that 4 and 8 have a common factor of 4, you

may remove that common factor from the numerator and denominator of the

product without writing the prime factorizations. However, make sure that the

numerator and denominator of the resulting fraction do not have any common

factors. If they do, continue to simplify.

Factor 8 as 2 4, and then remove the

common factors of 4 and 5 in the numerator

and denominator.

5

8



4

5



5  4

8  5



5

1

 4

1

2  4

1

 5

1



1

2

EXAMPLE 5

Multiply and simplify:

StrategyWe will multiply the numerators and denominators, and make sure that

the result is in simplest form.

WHYThis is the rule for multiplying three (or more) fractions.

Solution Recall from Section 2.4 that a product is positive when there are an

even number of negative factors. Since hastwonegative factors, the

product is positive.

To prepare to simplify, write 9, 14, and 10 in

prime-factored form.

Multiply the remaining factors in the numerator.

 (^) Multiply the remaining factors in the denominator.

3

10

To simplify, remove the common

factors of 2, 3, and 7 from the

numerator and denominator.



2

1

 3

1

 3  7

1

3

1

 2  7

1

 2

1

 5



2  3  3  7

3  2  7  2  5

Multiply the numerators.

Multiply the denominators.



2  9  7

3  14  10

Since the answer is positive,

drop both signs and continue.

2

3

a

9

14

ba

7

10

b

2

3

a

9

14

ba

7

10

b

321 ^14921 ^1072

a

7

10

a b

9

14

b

2

3

Self Check 5

Multiply and simplify:

Now TryProblem 37

2

5

a

15

22

ba

11

26

b

Caution! In Example 5, it was very helpful to prime factor and simplify when

we did (the third step of the solution). If, instead, you find the product of the

numerators and the product of the denominators, the resulting fraction is difficult

to simplify because the numerator, 126, and the denominator, 420, are large.

cc

Evaluate exponential expressions that have fractional bases.

We have evaluated exponential expressions that have whole-number bases and

integer bases. If the base of an exponential expression is a fraction, the exponent tells

us how many times to write that fraction as a factor. For example,

Since the exponent is 2, write the base,

as a factor 2 times.

2

a^23 ,

3

b

2



2

3



2

3



2  2

3  3



4

9

3

Don’t multiply in the numerator and

denominator and then try to simplify

the result. You will get the same

answer, but it takes much more work.

Factor and simplify at this

stage, before multiplying

in the numerator and

denominator.

2

3



9

14



7

10



2  9  7

3  14  10



126

420