INDEX LAWSProof and tutorial

Index Laws

Learning Objectives:

• Able to estimate the answer to a question involving indices

• Able to multiply terms with the same base

• Able to divide terms with the same base

Index Laws• How can I simplify: 45 x 43?

• How can I simplify: 79 x 76?

4 x 4 x 4 x 4 x 4

4 x 4 x 4x = 48

7 x 7 x 7 x 7 x 7 x 7 x 7

7 x 7 x 7 x 7 x 7 x 7

x = 715

Law 1 : Multiplication

26 x 24 = 210

24 x 22 = 26

35 x 37 = 312

General Rule:

When multiplying terms with the same base, you add the powers

am x an = am+n

EXAMPLE: 47 x 45 = 47+5 = 412

Index Laws• How can I simplify 95 ÷ 93?

• How can I simplify 117 ÷ 112?

9 x 9 x 9 x 9 x 99 x 9 x 9

= 9 x 9 = 92

11 x 11 x 11 x 11 x 11 x 11 x 11 11 x 11

= 115

Law 2 : Division

26 ÷ 24 = 22

25 ÷ 22 = 23

35 ÷ 37 = 3-2

General Rule

When dividing terms with the same base, you subtract the powers

am ÷ an = am-n

EXAMPLE: 47 ÷ 45 = 47-5 = 42

Index Laws

• How can I simplify (43)4?

• How can I simplify (72)5?

4 x 4 x 4

4 x 4 x 4

4 x 4 x 4

4 x 4 x 4

x x x = 412

7 x 7 7 x 77 x 77 x 77 x 7x xxx = 710

Law 3 : Brackets

(26)2 = 26 x 26 = 212

(35)3 = 35 x 35 x 35 = 315

General Rule:

When raising a power to another power, you multiply the powers

(am)n = am x n EXAMPLE: (47)5= 47 x 5 = 435

Index Laws• Simplify the following:

73 x 78

75

4 x 5 x 10

Index Laws

Learning Objectives:

• Able to multiply terms with the same base• Able to divide terms with the same base• Able to simplify terms with negative

indices

Level 8 13/04/2023

Index Laws• How can I simplify 5a3 x 6a7?

5 x a x a x a 6 x a x a x a x a x a x a x a

x

5 x 6 x a x a x a x a x a x a x a x a x a x a

= 30a10

Algebra and Indices• 9d6 x 8d3

• 12e10 x 6e4

• 24f8 ÷ 6f4

• 60g12 ÷ 5g3

Starter

• 89 ÷ 83

• (35)8

• 47 x 76

• (2y3)4

Index Laws

Learning Objectives:

• Able to multiply and divide terms with the same base

• Able to raise a power to another power

• Able to simplify terms with negative

indices

Algebra and Indices• (7c3)2 =

= 7 x 7 x c x c x c x c x c x c

= 49c6

• (4j8)3

• (5k-3)4

7 x c x c x cx 7 x c x c x c

Law 4: Negative Indices25 = 3224 = 1623 = 822 = 421 = 220 = 12-1 = ½ 2-2 = ¼ 2-3 = ⅛2-4 = 116

General Rule

a-n = 1an

20 =2-1 =2-2 =2-3 = 2-4 =

Law 5: Power 025 = 3224 = 1623 = 822 = 421 = 220 = 12-1 = ½ 2-2 = ¼ 2-3 = ⅛2-4 = 116

GENERAL RULE:

Any term to the power 0 is equal to 1.

Negative Indices• 6-2

• 7-3

• 9m-4

=

= =

=

= 9 x =

1. (38)4 2. 7f3 x 9f9

3. 124 x 1210

4. (6m5)3

5. 25 ÷ 2-3 6. 42g7 ÷ 7g2

7. 39 x 36

358. 116 x 116

112 x 113

9. 7-2 10. 3-3

Simplify the following, leaving your answers in index notation.

Solve the following.

= 332 = 63f12

= 1214 = 216m15

= 28 = 6g5

= 310 = 117

= =

Negative Indices

Simplify:

a) b) c)

Solve:

d) e) f)

Simplify:

g) h) i)

ANSWERS:

a) b) c)

d) e) f)

g) h) i)

More difficult examples• Simplify:

• Solve:

43 x 47

46

6p8 x 3p3

9p4 x p7

83

85 (4-1)3