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Laws of exponents

Exponential notation is a short way of writing the same number multiplied by itself many times. We will now have a closer look at writing numbers using exponential notation. Exponents can also be called indices.

Image

For any real number a and natural number n, we can write a multiplied by itself n times as an.

Identity 1

  1. an=a×a×a××a(ntimes)(a,n)

  2. a0=1 (a0 because 00 is undefined)

  3. a-n=1an (a0 because 10 is undefined)

Examples:

  1. 3×3=32

  2. 5×5×5×5=54

  3. p×p×p=p3

  4. (3x)0=1

  5. 2-4=124=116

  6. 15-x=5x

Notice that we always write the final answer with positive exponents.

Chapter introduction

Laws of exponents

There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been done in earlier grades, but we list all the laws here for easy reference:

Identity 2

  • am×an=am+n

  • aman=am-n

  • (ab)n=anbn

  • abn=anbn

  • (am)n=amn

where a>0, b>0 and m,n.

Example 1: Applying the exponential laws

Question

Simplify:

  1. 23x×24x

  2. 12p2t53pt3

  3. (3x)2

  4. (3452)3

Answer

  1. 23x×24x=23x+4x=27x

  2. 12p2t53pt3=4p(2-1)t(5-3)=4pt2

  3. (3x)2=32x2=9x2

  4. (34×52)3=3(4×3)×5(2×3)=312×56

Example 2: Exponential expressions

Question

Simplify: 22n×4n×216n

Answer

Change the bases to prime numbers
22n×4n×216n=22n×(22)n×21(24)n(1)
Simplify the exponents
=22n×22n×2124n=22n+2n+124n=24n+124n=24n+1-(4n)=2(2)

Example 3: Exponential expressions

Question

Simplify: 52x-19x-2152x-3

Answer

Change the bases to prime numbers
52x-19x-2152x-3=52x-1(32)x-2(5×3)2x-3=52x-132x-452x-332x-3(3)
Subtract the exponents (same base)
=5(2x-1)-(2x-3)×3(2x-4)-(2x-3)=52x-1-2x+3×32x-4-2x+3=52×3-1(4)
Write the answer as a fraction
=253=813(5)

Important: when working with exponents, all the laws of operation for algebra apply.

Example 4: Simplifying by taking out a common factor

Question

Simplify: 2t-2t-23×2t-2t

Answer

Simplify to a form that can be factorised
2t-2t-23×2t-2t=2t-(2t×2-2)3×2t-2t(6)
Take out a common factor
2t-2t-23×2t-2t=2t(1-2-2)2t(3-1)(7)
Cancel the common factor and simplify
2t-2t-23.2t-2t=1-142=342=38(8)

Example 5: Simplifying using difference of two squares

Question

Simplify: 9x-13x+1

Answer

Change the bases to prime numbers
9x-13x+1=(32)x-13x+1=(3x)2-13x+1(9)
Factorise using the difference of squares
9x-13x+1=(3x-1)(3x+1)3x+1(10)
Simplify
9x-13x+1=3x-1(11)

Exercise 1:

Simplify without using a calculator:

  1. 160

  2. 16a0

  3. 2-232

  4. 52-3

  5. 23-3

  6. x2x3t+1

  7. 3×32a×32

  8. a3xax

  9. 32p24p8

  10. (2t4)3

  11. (3n+3)2

  12. 3n9n-327n-1

1. 160=1

 

2.  16a0=16(1)=16

 

3. 2-232=1223.2=14×9=136

 

4. 52-3=(5)(23)=(5)(8)=40

 

5. (23)-3=2-33-3=3323=278

 

6. x2.x3t+1=x2.x3t.x1=x2+1.x3t=x3x3t

 

7. 3×32a×32=31+2a+2=32a+3

 

8. a3xax=a3x.a-x=a3x-x=a2x

 

9. 32p24p8=8p2-8=8p-6=8p6

 

10. (2t4)3=23.t4.3=8t12

 

11. (3n+3)2=32n+6

 

12. 3n.9n-327n-1=3n.(32)n-333n-1=3n.32n-633n-3=3n+2n-6-(3n-3)=33n-6-3n+3=3-3=133=127