Transcript Index laws 1

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Index notation
We use index notation to show repeated
multiplication by the same number.
For example:
We can use index notation to write 2 × 2 × 2 × 2 × 2 as:
index or power
25
base
This number is read as ‘two to the power of five’.
25 = 2 × 2 × 2 × 2 × 2 = 32
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The first index law
When we multiply two numbers written in index form and with
the same base we can see an interesting result.
For example:
34 × 32 = (3 × 3 × 3 × 3) × (3 × 3)
=3×3×3×3×3×3
= 36 = 3(4 + 2)
73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7)
=7×7×7×7×7×7×7×7
= 78 = 7(3 + 5)
When we multiply
numbers
with the same
What two
do you
notice?
base the indices are added.
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The second index law
When we divide two numbers written in index form and with
the same base we can see another interesting result.
For example:
45
÷
42
4×4×4×4×4
=
= 4 × 4 × 4 = 43 = 4(5 – 2)
4×4
56 ÷ 54 =
5×5×5×5×5×5
= 5 × 5 = 52 = 5(6 – 4)
5×5×5×5
When we divide
with the same
Whattwo
do numbers
you notice?
base the indices are subtracted.
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Zero indices
Look at the following division:
64 ÷ 64 = 1
Using the second index law:
64 ÷ 64 = 6(4 – 4) = 60
This means that:
60 = 1
In fact, any number raised to the power of 0 is equal to 1.
For example:
100 = 1
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3.4520 = 1
723 538 5920 = 1
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Negative indices
Look at the following division:
3×3
3×3×3×3
32 ÷ 34 =
=
1
1
= 2
3×3
3
Using the second index law:
32 ÷ 34 = 3(2 – 4) = 3–2
That means that:
3–2 =
Similarly:
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6–1 =
1
6
7–4 =
1
32
1
74
and
5–3 =
1
53
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Using algebra
We can write all of these results algebraically.
am × an = a(m + n)
am ÷ an = a(m – n)
a1 = a
a0 = 1
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a–1 =
1
a
a–n =
1
an
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Using index laws
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