Basic Engineering Mathematics, Fifth Edition

52 Basic Engineering Mathematics

Dividing each term by the HCF (i.e. 2^2 )gives

28 × 3 −^4

22 × 33 − 23

=

26 × 3 −^4

33 − 2

=

26

34 ( 33 − 2 )

Problem 23. Simplify

(

4

3

) 3

×

(

3

5

)− 2

(

2

5

)− 3 giving

the answer with positive indices

Raisinga fractionto a power means that boththe numer-

ator and the denominator of the fractionare raised to that

power, i.e.

(

4

3

) 3

=

43

33

A fraction raised to a negative power has the same value

as the inverse of the fraction raised to a positive power.

Thus,

(

3

5

)− 2

=

1

(

3

5

) 2 =

1

32

52

= 1 ×

52

32

=

52

32

Similarly,

(

2

5

)− 3

=

(

5

2

) 3

=

53

23

Thus,

(

4

3

) 3

×

(

3

5

)− 2

(

2

5

)− 3 =

43

33

×

52

32

53

23

=

43

33

×

52

32

×

23

53

=

( 22 )^3 × 23

3 (^3 +^2 )× 5 (^3 −^2 )

=

29

35 × 5

Now try the following Practice Exercise

PracticeExercise 31 Further problemson

indices (answers on page 342)

In problems 1 to 4, simplify the expressions given,

expressing the answers in index form and with

positive indices.

1.

33 × 52

54 × 34

2.

7 −^2 × 3 −^2

35 × 74 × 7 −^3

3.

42 × 93

83 × 34

4.

8 −^2 × 52 × 3 −^4

252 × 24 × 9 −^2

InProblems5to15,evaluatetheexpressionsgiven.

5.

(

1

32

)− 1

6. 81^0.^25


7. 16

−

1

(^4) 8.
(
4
9
) 1 / 2
9.
92 × 74
34 × 74 + 33 × 72
10.
33 × 52
23 × 32 − 82 × 9
11.
33 × 72 − 52 × 73
32 × 5 × 72
12.
( 24 )^2 − 3 −^2 × 44
23 × 162
13.
(
1
2
) 3
−
(
2
3
)− 2
(
3
2
) 2 14.
(
4
3
) 4
(
2
9
) 2
15.
( 32 )^3 /^2 ×( 81 /^3 )^2
( 3 )^2 ×( 43 )^1 /^2 ×( 9 )−^1 /^2