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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

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

This leads to =a

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-

Page 1 of 3

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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

This leads to =a

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-

Page 1 of 3

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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

Page 2 of 3

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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

Page 2 of 3

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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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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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