The LessonThe 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 ExponentsLet's start with the basic laws. These are special cases of a base with an exponent.
|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 PowersWhen multiplying the same letter with exponents, add the exponents.
Example: a2 × a3 = a2 + 3 = a5Read more about multiplying powers in algebra
Dividing PowersWhen dividing the same letter with exponents, subtract the exponents.
Example: a5 ÷ a3 = a5 - 3 = a2Read more about dividing powers in algbra
Powers of a PowerWhen raising one exponent to another, multiply the exponents.
Example: (a2)3 = a2 × 3 = a6Read more about finding a power of a power in algebra
Power of an Algebraic FractionWhen raising a fraction to an exponent, raise both the numerator and denominator to the exponent.
Example: (a ⁄ b)2 = a2 ⁄ b2Read more about finding the power of an algebraic fraction
Exponent Is NegativeA negative exponent means calculating the positive exponent and finding the reciprocal (i.e. find 1 over it).
Example: a−2 = 1 ⁄ a2Read 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 nth root of the base. n = 2 is the square root.
n = 3 is the cube root.
Example: a½ = √aRead more about how to find a fractional exponent in algebra
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: a3⁄2 = √(a3) = (√a)3Read more about how to find a fractional exponent in algebra
Lesson SlidesThe laws of exponents in algebra are often not used in isolation of each other, but are needed together. The slider below shows real examples of how to use the laws of exponents. Open the slider in a new tab
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, a2 means that a is multiplied by itself 2 times:
a2 = a × a
There Are No Rules for Adding or Subtracting ExponentsThere are no rules for adding or subtracting exponents. They just stay as they are:
Mathematics Monster has known some students who have got confused with other laws of exponents and have made up their own rules:
The correct rules are:
Just the exponents are added or subtracted.