2. INDICES
2
> The index refers to the power to which a number is raised.
> In the example 5³, the number 5 is raised to the power of 3, which
means 5 x 5 x 5.
> The 3 is known as the index and the 5 is known as the base.
> Indices is the plural form of index.
Here are some examples:
> 33 = 3 x 3 x 3 = 27
> 74 = 7 x 7 x 7 x 7 = 2401
> 31 = 3
3. LAWS OF INDICES
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When working with numbers involving indices there are
three basic laws which can be applied. These are:
1. am x an = am+n
2. am ÷ an = am-n
3. (am)n = amxn
4. Worked examples
a. Simplify 43 x 44.
43 x 44 = 4(3+4) = 47
b. Evaluate (42)3.
(42)3 = 42x3 = 46 = 4096
c. Simplify 2 x 2 x 2 x 5 x 5 using indices.
2 x 2 x 2 x 5 x 5 = 23 x 52
a. Write out 43 x 63 in full.
43 x 63 = 4 x 4 x 4 x 6 x 6 x 6
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5. The Zero Index
5
> The zero index indicates
that a number is raised
to the power 0.
> A number that is raised
to the power 0 is equal to
1.
am ÷ an = am-n therefore
= a0
However,
therefore a0 = 1
m
m
m
m
a
a
a
1
m
m
a
a
6. Negative Indices
6
> A negative index
indicates that a number
is being raised to a
negative power: e.g 4-3.
> Another law of indices
states that
This can be proved as follows:
a-m = a0-m
=
=
therefore
m
a
a0
m
m
a
a
1
m
m
a
a
1
m
a
1
7. Exponential equations
> Equations that involve indices as unknown are known as
exponential equations.
> Worked examples:
a. Find the value of x if 2x = 32
32 can be expressed as a power of 2, 32 = 25
Therefore 2x = 25 -> x = 5
b. Find the value of m if 3(m-1) = 81
81 can be expressed as a power of 3, 81 = 34
Therefore 3(m-1) = 34 m – 1 = 4
m = 5
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8. STANDARD FORM
> Standard form is also known as standard index form or
sometimes as scientific notation.
> Positive indices and large numbers
- 100 = 1 x 102
- 1000 = 1 x 103
- 10000 = 1 x 104
- 3000 = 3 x 103
- For a number to be in standard form it must take the form
A x 10n where the index n is a positive or negative integer
and A must lie in the range 1 ≤ A < 10.
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9. Worked examples
a. Write 72000 in standard form.
72000 = 7.2 x 104
b. Write 4 x 104 as an ordinary number.
4 x 104 = 4 x 10000 = 40000
c. Multiply the following and write your answer in standard form.
600 x 4000 = 2400000 = 2.4 x 106
d. Multiply the following and write your answer in standard form.
(2.4 x 104) x ( 5 x 107) = 12 x 1011
= 1.2 x 1012 (standard form)
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10. Worked examples
e. Divide the following and write your answer in standard form:
(6.4 x 107) ÷ (1.6 x 103) = 4 x 104
f. Add the following and write your answer in standard form:
(3.8 x 106) + (8.7 x 104)
Changing the indices to the same value gives the sum:
(380 x 104) + (8.7 x 104) = 388.7 x 104
= 3.887 x 106 in standard form
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11. Worked examples
g. Subtract the following and write your answer in standard form:
(6.5 x 107) – (9.2 x 105)
Changing the indices to the same value gives the sum:
(650 x 105) – (9.2 x 105) = 640.8 x 105
= 6.408 x 107 in standard form
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12. NEGATIVE INDICES AND SMALL NUMBERS
> A negative index is used when writing a number between 0
and 1 in standard form.
- 100 = 1 x 102
- 10 = 1 x 101
- 1 = 1 x 100
- 0.1 = 1 x 10-1
- 0.01 = 1 x 10-2
- 0.001 = 1 x 10-3
- 0.0001 = 1 x 10-4
- Note that A must still lie in the range 1 ≤ A < 10.
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13. Worked examples
a. Write 0.0032 in standard form.
0.0032 = 3.2 x 10-3
b. Write 1.8 x 10-4 as an ordinary number.
1.8 x 10-4 = 1.8 ÷ 104
= 1.8 ÷ 10000
= 0.00018
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14. Worked examples
c. Write the following numbers in order of magnitude, starting
from the largest.
3.6 x 10-3 5.2 x 10-5 1 x 10-2 8.35 x 10-2 6.08 x 10-8
Answer:
8.35 x 10-2 1 x 10-2 3.6 x 10-3 5.2 x 10-5 6.08 x 10-8
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15. Fractional indices
= 41
= 4
> Therefore
but
> therefore
15
.
)
4
(
16 2
1
2
2
1
as
written
be
can
)
2
1
2
(
2
1
2
4
)
4
(
4
162
1
16
162
1
4
16
17. Worked examples:
a. Evaluate without the use of a calculator.
Alternatively:
= 21
= 2 = 2
17
4
4
1
16
16
4
1
16
4 4
2
4
1
4
4
1
)
2
(
16
18. Worked examples:
b. Evaluate without the use of a calculator.
Alternatively:
= 53
= 53 = 125
= 125
18
3
2
1
2
3
)
25
(
25
2
3
25
3
)
25
(
2
3
2
2
3
)
5
(
25
19. Worked examples
c. Solve 32x = 2
32 is 25 so
or
therefore
d. Solve 125x = 5
125 is 53 so
or
therefore
19
2
32
5
2
325
1
5
1
x
5
125
3
5
1253
1
3
1
x