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
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