# Negative Indices

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This pdf includes the following topics:-
Negative indices
Reciprocal
Integers raised to negative powers
Fractions raised to negative power
Negative Indices involving algebra
Examples
1. www.mathspanda.com
Negative indices
Starte
1 (Review of last lesson)
Simplify (a 1250 (b 8x 6 × 5x 9 (c (a 3)4 ÷ a 9
2 (Review of last lesson)
Find the value of x in the following expressions
(a ax ÷ a7 = a9 (b (63x )4 = 648
2 With a partner, decide whether the answer of the following is A, B, C or D. Justify your
answer with working or an explanation
1
2−3 A. −6 B. −8 C. D. −1
8
Notes
1
Negative indices (“one over…”) a −1 =
a
1
a −n = n
a
1
a −1
1
The word reciprocal is often associated with negative indices because the reciprocal of 5 is .
5
The reciprocal of a number or expression is “one over” the number or expression
Integers raised to negative power
E.g. 1 Without using a calculator, simplify the following
1 1
(a 7−1 (b (c 5−3 (d
9−1 6−2
1
Working: (a 7−1 =
7
Exercis
9-1 class textbook p42 M2.7 Qu 1, 2ace…, 3-
A*-G class textbook p39 E2.1 Qu 1, 2ace…, 3-
9-1 homework book p13 M2.7 Qu 1-
A*-G homework book p10 E2.1 Qu 1-
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2. www.mathspanda.com
Fractions raised to negative power
E.g. 2 By using division by a fraction, and without a calculator, state the value of
(4) (3)
−1 −1
1 2
(a (b
(4)
−1
1 1
Working: (a = 1
negative indices ≡ “one over…”
4
1
=1÷ dividing by fraction
4
4
=1× ip fraction and multiply
1
=4
When a fraction is raised to a negative power, “ ip” the fraction and make the power positive
(b) (a )
−n n
a b
i.e. =
E.g. 3 Without a calculator, nd the value of
( 12 ) (8) (3) (5)
−1 −1 −2 −2
1 7 1 3
(a (b (c (d
( 12 )
−1
1 12
Working: (a = = 12
1
(8)
−1
7 8
(b =
7
(3) (1 )
−2 2
1 3
(c = = 32 = 9
(5) (3 )
−2 2
3 5 52 25
(d = = 2 =
3 9
N.B. If a mixed number is raised to a negative power, turn it into an improper fraction and
then nd the negative reciprocal
E.g. 4 Without a calculator, nd the value of
( 5) ( 4)
−3 −2
1 1
(a 1 (b 2
( 5) (5 ) (6 )
−3 −3 3
1 6 5 53 125
Working: (a 1 = = = 3 =
6 216
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3. www.mathspanda.com
Exercis
9-1 class textbook p43 M2.7 Qu 7,
A*-G class textbook p40 M2.7 Qu 6,
9-1 homework book p13 M2.7 Qu 5-
A*-G homework book p11 E2.1 Qu 5,
Negative Indices involving algebr
1
E.g. 5 Simplify (a × a7 (b g 6 ÷ g −6 (c (b −5)−1
a 10
1 7 −10 7 −3 1
Working: (a × a = a × a = a =
a 10 a3
Video Negative indices
Solutions to Starter and E.g.s
Exercis
9-1 class textbook p42 M2.7 Qu 9-1
A*-G class textbook p37 E2.1 Qu 8-1
9-1 homework book p13 M2.7 Qu 8-1
A*-G homework book p10 E2.1 Qu
Summar
1 1 1
Negative indices (“one over…”) a −1 = =a a −n =
a −1a an
1
The word reciprocal is often associated with negative indices because the reciprocal of 5 is .
5
(b) (a )
−n n
a b
=
If a mixed number is raised to a negative power, turn it into an improper fraction and then nd
the negative reciprocal
Page 3 of 3
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