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Exponents


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Numeracy Basics

Exponents

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This topic will cover how to:

  • Evaluate exponents and write repeated multiplication using exponents; and
  • Apply the ten exponent laws to rearrange and simplify 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

= 3 x 3
= 9
= 8 x 8 x 8
= 512
= 24
= 105 x 22

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

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

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

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of Exponents.

 


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