The Lesson
The laws of exponents are rules for using exponents in algebra. Imagine you see a letter that has an exponent. For example, the letter a with an exponent of 2. In this example, a (called the base) is multiplied by itself 2 (the exponent) times. What if we see the a letter with an exponent multiplying that same letter 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 letter with exponents, add the exponents.
Example: a^{2} × a^{3} = a^{2 + 3} = a^{5}
Read more about multiplying powers in algebra
Dividing Powers
When dividing the same letter with exponents, subtract the exponents.
Example: a^{5} ÷ a^{3} = a^{5 - 3} = a^{2}
Read more about dividing powers in algbra
Powers of a Power
When raising one exponent to another, multiply the exponents.
Example: (a^{2})^{3} = a^{2 × 3} = a^{6}
Read more about finding a power of a power in algebra
Power of an Algebraic Fraction
When raising a fraction to an exponent, raise both the numerator and denominator to the exponent.
Example: (a ⁄ b)^{2} = a^{2} ⁄ b^{2}
Read more about finding the power of an algebraic fraction
Exponent Is Negative
A negative exponent means calculating the positive exponent and finding the reciprocal (i.e. find 1 over it).
Example: a^{−2} = 1 ⁄ a^{2}
Read more about negative exponents in algebra
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: a^{½} = √a
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: a^{3⁄2} = √(a^{3}) = (√a)^{3}
What Is an Exponent?
An exponent tells you how many times a number or letter 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, a^{2} means that a is multiplied by itself 2 times:
a^{2} = a × a