The Laws of Exponents in Algebra
(KS3, Year 7)
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 | 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 Powers
When multiplying the same letter with exponents, add the exponents.
Example: a2 × a3 = a2 + 3 = a5
multiplying powers in algebra
Dividing Powers
When dividing the same letter with exponents, subtract the exponents.
Example: a5 ÷ a3 = a5 - 3 = a2
dividing powers in algbra
Powers of a Power
When raising one exponent to another, multiply the exponents.
Example: (a2)3 = a2 × 3 = a6
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 = a2 ⁄ b2
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 ⁄ a2
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 = √a
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)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, a2 means that a is multiplied by itself 2 times:
a2 = a × a
Beware
There Are No Rules for Adding or Subtracting Exponents
There 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.Worksheet
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