The Laws of Exponents
The Laws of Exponents
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). 
Read more about finding an exponent of −1 in algebra
Let's look at the more complicated laws of exponents.
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
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