The Lesson
The laws of exponents are rules for using exponents. An exponent is a small, raised number written to the right side of another number. For example, the number 2 with an exponent of 2 is shown below: An exponent tells you how many times a number is multiplied by itself. In this example, 2 (called the base) is multiplied by itself 2 (the exponent) times. What if we see the a number with an exponent multiplying that same number with a different exponent? Or dividing? What if the exponent is negative? Or a fraction? We need to know the laws of exponents.The Laws of Exponents
Let's start with the basic laws. These are special cases of a base with an exponent.Law | Explanation | |
---|---|---|
Base of 1 | 14 = 1 × 1 × 1 × 1 = 1 | |
Exponent of 0 | Any base with an exponent of 0 is 1. | |
Exponent of 1 | Any base with an exponent of 1 is equal to the base. | |
Exponent of −1 | Any base with an exponent of −1 is equal to 1 divided by the base (the reciprocal of the base). |
Multiplying Powers
When multiplying the same number with exponents, add the exponents.Example: 22 × 23 = 22 + 3 = 25
25 = 2 × 2 × 2 × 2 × 2 = 32
Dividing Powers
When dividing the same number with exponents, subtract the exponents.Example: 25 ÷ 23 = 25 - 3 = 22
22 = 2 × 2 = 4
Powers of a Power
When raising one exponent to another, multiply the exponents.Example: (22)3 = 22 × 3 = 26
26 = 2 × 2 × 2 × 2 × 2 × 2 = 64
Power of a Fraction
When raising a fraction to an exponent, raise both the numerator and denominator to the exponent.Example: (2 ⁄ 3)2 = 22 ⁄ 32
22 ⁄ 32 = (2 × 2) ⁄ (3 × 3) = 4 ⁄ 9
Exponent Is Negative
A negative exponent means calculating the positive exponent and finding the reciprocal (i.e. find 1 over it).
Example: 2-2 = 1 ⁄ (2 × 2) = 1 ⁄ 4
negative exponents
Exponent Is a Fraction (Numerator is 1)
A fractional exponent (where the fraction is 1 over n) means finding the nth root of the base. n = 2 is the square root.n = 3 is the cube root.
Example: 2½ = √2
Exponent Is a Fraction (Numerator is not 1)
To find a fractional exponent (where the fraction is m over n), either:- Find the mth power, and take the nth root, or
- Take the nth root, and find the mth power.
Example: 23⁄2 = √(23) = √(2 × 2 × 2) = √8 or
(√2)3 = √2 × √2 × √2 = √8
What Is an Exponent?
An exponent tells you how many times a number (or other quantity) is multiplied by itself. An exponent is denoted by a raised number by the right hand side of the number (called the base) that is multiplied by itself. For example, 32 means that 3 is multiplied by itself 2 times:
32 = 3 × 3 = 9