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Index Laws
Objectives:
A Grade
Use Index Notation and Index Laws for
Fractional Powers such as 16
A* Grade
Use Index Notation and Index Laws for
Fractional Powers such as 16
1
2
3
4
Prior knowledge:
Understand :
Use Index Laws for positive and negative powers
Index Laws
Using the index laws to simplify
a3 × a4
a3+4 = a7
Understanding that we add the index numbers we can
deduce the meaning of:
100½ × 100½
By adding the index numbers
100½+½ = 1001 = 100
Therefore 100½ is a number, that, when multiplied by itself is 100.
We know that:
√100 × √100 = 100
Therefore:
100½ = √100
The general rule is:
x½ = √x
Index Laws
Understanding that we add the index numbers we can
deduce the meaning of:
1
1
1
x 3 × x 3 × x3
1 1
1
By adding the index numbers
+
+
100 3 3 3 = 1001 = 100
3√100
Therefore:
1
3
100 =
3√100
× 3 √100 × 3 √100 = 100
x
The rule is:
1
a
1
3
= 3√x
The general rule is: x = a√x
Index Laws
Now do these:
1.
2.
Evaluate
a) 4½ 2
1
3
1
3
b) 40 1
c) 8 2
d) 125 5
1
5
1
3
1
5
Evaluate
a) 32
1
5
2
b) 243
3
c) 512
8
d) 59 049 9
Index Laws
More complex fractional index laws together:
(73) × (73) = 73+3 = 76
(73)2 means
(4½)3 means
4½ × 4½ × 4½
Therefore when we see 4
=4
3
2
3
2
We know it means (4½)3
In general a
y
x
1
x
actually means (4 )y
Index Laws
Now do these:
1.
Evaluate
a) 4
2.
a) 64
3
2
8
b) 27
2
3
3
4
9
c) 16 8
2
3
d) 125 25
Evaluate
5
6
32
b) 243
3
5
81 c) 512
2
3
64
3
4
d) 6561 729
Index Laws
Now do these:
3.
Evaluate
2
3
a) -8
b) (-125)
8
e) 256 -0.25
3
2
2
3
1
1
-0.25
f) 16
2
4

h) 49 × 81
1
4

25
c) 4
2
3
1
2
1
2
d) 16
2
3
g) 125 × 8
100
1

1
114
i) 5-2 × 105 × 16 2 1000
3
-0.5
1
4