Exponents
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Numeracy Basics
Exponents
Exponents
Exponents, otherwise known as indices or powers, are used as a shorthand notation for repeated multiplication. For example:
5 x 5 x 5 x 5 = 54
In this case we say that the number 5 is the base and the number 4 is the exponent or power.
Some other examples of exponents, and how they are evaluated, are as follows:
22 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8
24 = 2 x 2 x 2 x 2 = 16
5 x 5 x 5 x 5 =
54
22 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8
24 = 2 x 2 x 2 x 2 = 16
Examples: Exponents
1. Evaluate:
a. 32 = 3 x 3
= 9
b. 83 = 8 x 8 x 8
= 512
2. Rewrite the following:
a. 2 x 2 x 2 x 2 = 24
b. 10 x 10 x 10 x 10 x 10 x 2 x 2 = 105 x 22
Activity 1: Practice Questions

The Exponent Laws
There are ten 'laws' you need to abide by when working with exponents.
When detailing these laws, we will use the variables x and/or y to represent any whole number bases and the variables a and/or b to represent any integer exponents. For example:
xa means x multiplied by itself a times
If x = 3 and a = 4 then xa = 34 = 3 x 3 x 3 x 3
x = 10 and a = 5 then xa = 105 = 10 x 10 x 10 x 10 x 10

xa means x multiplied by itself a times
If x = 3 and a = 4 then xa = 34 = 3 x 3 x 3 x 3
x = 10 and a = 5 then xa = 105 = 10 x 10 x 10 x 10 x 10
Exponent Laws 1 and 2
Now that we have defined our variables let's look at Exponent Laws 1 and 2, which are the simplest two index laws:
Law 1: x 1 = x
For example, 5 1 = 5
Law 2: x 0 = 1
For example, 4 0 = 1
Law 1: x 1 = x
For example, 5 1 = 5
Law 2: x 0 = 1
For example, 4 0 = 1
Exponent Laws 3 and 4
Exponent Laws 3 and 4 cover how to multiply and divide with exponents respectively, and are as follows:
Law 3: xaxb = xa+b

For example, 2324 = (2 x 2 x 2) x (2 x 2 x 2 x 2)
= 2 x 2 x 2 x 2 x 2 x 2 x 2
= 27
= 23+4
Law 4: xa/xb = xa-b

For example, 105/103 = 10x10x10x10x10/10x10x10
= 10 x 10
= 102
= 105-3
Law 3: xaxb = xa+b
For example, 2324 =
(2 x 2 x 2) x (2 x 2 x 2 x 2)
= 2 x 2 x 2 x 2 x 2 x 2 x 2
= 27
= 23+4
Law 4: xa/xb = xa-b
For example, 105/103 =
10x10x10x10x10/10x10x10
= 10 x 10
= 102
= 105-3
Examples: Exponent Laws 1 to 4
1.323335 = 32+3+5
= 310
2.5354/56 = 53+4/56
= 57/56
= 57-6
= 51
= 5
3.434-3 = 43+(-3)
= 43-3
= 40
= 1
= 32+3+5
= 310
= 53+4/56
= 57/56
= 57-6
= 51
= 5
= 43+(-3)
= 43-3
= 40
= 1
Activity 2: Practice Questions

Exponent Laws 5 and 6
Exponent Laws 5 and 6 cover negative exponents and exponents of exponents respectively, and are as follows:
Law 5: x -a= 1/x a
For example, 2 -3 = 1/2 3
For example, 4 -1 = 1/4 1 = 1/4
Law 6: (x a) b = x ab
For example, (2 2) 3 = (2 x 2) 3
= (2 x 2) x (2 x 2) x (2 x 2)
= 2 x 2 x 2 x 2 x 2 x 2
= 2 6
= 2 2x3
Law 5: x -a= 1/x a
For example, 2 -3 = 1/2 3
For example, 4 -1 = 1/4 1 = 1/4
Law 6: (x a) b = x ab
For example, (2 2) 3 =
(2 x 2) 3
= (2 x 2) x (2 x 2) x (2 x 2)
= 2 x 2 x 2 x 2 x 2 x 2
= 2 6
= 2 2x3
Exponent Laws 7 and 8
Exponent Laws 7 and 8 cover how to simplify when we have two bases and one exponent, and are as follows:
Law 7: (xy) a= x ay a
For example, (4 x 5) 3 = (4 x 5) x (4 x 5) x (4 x 5)
= 4 x 5 x 4 x 5 x 4 x 5
= 4 x 4 x 4 x 5 x 5 x 5
= 4 35 3
Law 8: (x/y) a= x a/y a
For example, (4/5) 3= 4/5 x 4/5 x 4/5
= 4 x 4 x 4/5 x 5 x 5
= 4 3/5 3
Law 7: (xy) a= x ay a
For example, (4 x 5) 3 =
(4 x 5) x (4 x 5) x (4 x 5)
= 4 x 5 x 4 x 5 x 4 x 5
= 4 x 4 x 4 x 5 x 5 x 5
= 4 35 3
Law 8: (x/y) a= x a/y a
For example, (4/5) 3=
4/5 x 4/5 x 4/5
= 4 x 4 x 4/5 x 5 x 5
= 4 3/5 3
Examples: Exponent Laws 5 to 8
1.(5-2)2 = 5-2x2
= 5-4
= 1/54
2.(3-44)-3 = (3 -4)-34-3
= 3-4x-34-3
= 3124-3
= 312
= 43
3.(5/6-2)3 = 53/(6-2)3
= 53/6-2x3
= 53/6-6
= 5366
= 5-2x2
= 5-4
= 1/54
= (3 -4)-34-3
= 3-4x-34-3
= 3124-3

= 53/(6-2)3
= 53/6-2x3
= 53/6-6
= 5366
Activity 3: Practice Questions

Exponent Laws 9 and 10
Exponent Laws 9 and 10 cover fractional exponents, and are as follows:
Law 9: x1/a = a√x
For example, 91/2 = 2√9 = √9
271/3 = 3√27
Law 10: x b/a = a√x b = ( a√x) b
For example, 8 2/3 = 8 2 x 1/3
= (8 2) 1/3 (from Law 6)
= 3√8 2 (from Law 9)
Similarly, 8 2/3 = 8 1/3 x 2
= (8 1/3) 2 (from Law 6)
=( 3√8 ) 2 (from Law 9)
Law 9: x1/a = a√x
For example, 91/2 =
2√9 =
√9
271/3 =
3√27
Law 10: x b/a = a√x b
= ( a√x) b
For example, 8 2/3 =
8 2 x 1/3
= (8 2) 1/3 (from Law 6)
= 3√8 2 (from Law 9)
Similarly, 8 2/3 = 8 1/3 x 2
= (8 1/3) 2 (from Law 6)
=( 3√8 ) 2 (from Law 9)
Examples: Exponent Laws 9 and 10
1.161/4 = 4√16
= 2
2.275/3 = (3√27)5
= 3 5
= 243
= 4√16
= 2
= (3√27)5
= 3 5
= 243
Activity 4: Practice Questions

End of Topic

Congratulations, you have completed this topic.
You should now have a better understanding of Exponents.
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