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Exponents

Three to the fourth power with the exponent of four written as a small number above and to the right of the base number three.

An exponent tells how many times to multiply a number (the ‘base’ number).

34 = (3)(3)(3)(3) = 81

xn = (x)(x)(x)…

(x is mulitpled n times)

Exponents are shorthand: it is much quicker to write 54 than (5)(5)(5)(5)

Examples:

32 = (3)(3) = 9
33 = (3)(3)(3) = 27
34 = (3)(3)(3)(3) = 81

Example: Solve 54

  1. Multiply the base by itself: (5)(5) = 25
  2. Multiply the product times the base (third time): (25)(5) = 125
  3. Multiply the product times the base (fourth time): (125)(5) =625

52 = (5)(5) = 25
53 = (5)(5)(5) = (25)(5) = 125
54 = (5)(5)(5)(5) = (125)(5) = 625
55 = (5)(5)(5)(5)(5) = (625)(5) = 3125

Pattern of Positive and Negative Exponents

A positive base raised to a positive number gets larger as the exponent gets higher.
An exponential term with a negative exponent gets smaller as the exponent gets further below zero.

Example: Pattern of Positive and Negative Exponents with Base 4
44 (4·4·4·4) 256  
43 (4·4·4) 64
42 (4·4) 16
41 4 4
40 1 1
4-1 141 14 14
4-2 142 1(4·4) 116
4-3 143 1(4·4·4) 164
4-4 144 1(4·4·4·4) 1256

Notes:
x0 = 1
(except if x is a 0. 00 has no answer.)

1xa is the same as x-a. A term with a negative exponent is equal to the reciprocal of the term.

Why do we say ‘Squared’ and ‘Cubed’?

42 can be read:

"Four to the power of two" - or -

"Four to the second power" - or -

"Four squared"

4 by 4 square

We say "four squared" for the exponent 2 because the area of a square is equal to the length of one of its sides to the power of two.

Area = width · height
Area = 4 · 4 or 42

43 can be read:

"Four to the power of three" - or -

"Four to the third power" - or -

"Four cubed"

cube with a volume of 4 by 4 by 4

We say "four cubed" for the exponent 3 because the volume of a cube is equal to the length of one of its edges to the power of three.

Volume = width · height · depth
Volume = 4 · 4 · 4 or 43

Two and three are the only exponents that are usually called by an additional name.

How are exponents useful?

Exponents are shorthand: it is much quicker to write 54 than (5)(5)(5)(5).

Exponents can represent change at a constant rate, for example:

  • A population doubling every year (e.g. population change rate = (p)(2x) where x = number of years)

Vocabulary

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