The Laws of Exponents
(KS3, Year 7)
The LessonThe 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 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 number with exponents, add the exponents.
Example: 22 × 23 = 22 + 3 = 25 25 = 2 × 2 × 2 × 2 × 2 = 32Read more about multiplying powers
Dividing PowersWhen dividing the same number with exponents, subtract the exponents.
Example: 25 ÷ 23 = 25 - 3 = 22 22 = 2 × 2 = 4Read more about dividing powers
Powers of a PowerWhen raising one exponent to another, multiply the exponents.
Example: (22)3 = 22 × 3 = 26 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64Read more about finding a power of a power
Power of a FractionWhen 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 ⁄ 9Read more about finding the power of fraction
Exponent Is NegativeA negative exponent means calculating the positive exponent and finding the reciprocal (i.e. find 1 over it).
Example: 2-2 = 1 ⁄ (2 × 2) = 1 ⁄ 4Read more about 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
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 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
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.
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