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 | 1^{4} = 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: 2^{2} × 2^{3} = 2^{2 + 3} = 2^{5}
2^{5} = 2 × 2 × 2 × 2 × 2 = 32
Dividing Powers
When dividing the same number with exponents, subtract the exponents.Example: 2^{5} ÷ 2^{3} = 2^{5 - 3} = 2^{2}
2^{2} = 2 × 2 = 4
Powers of a Power
When raising one exponent to another, multiply the exponents.Example: (2^{2})^{3} = 2^{2 × 3} = 2^{6}
2^{6} = 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} = 2^{2} ⁄ 3^{2}
2^{2} ⁄ 3^{2} = (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
Read more about negative exponents
Exponent Is a Fraction (Numerator is 1)
A fractional exponent (where the fraction is 1 over n) means finding the n^{th} root of the base. n = 2 is the square root.n = 3 is the cube root.
Example: 2^{½} = √2
Exponent Is a Fraction (Numerator is not 1)
To find a fractional exponent (where the fraction is m over n), either:- Find the m^{th} power, and take the n^{th} root, or
- Take the n^{th} root, and find the m^{th} power.
Example: 2^{3⁄2} = √(2^{3}) = √(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, 3^{2} means that 3 is multiplied by itself 2 times:
3^{2} = 3 × 3 = 9