The Laws of Exponents
(KS3, Year 7)

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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: 2 squared

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. 2 squared equals 2 times 2

What if we see the a number with an exponent multiplying that same number with a different exponent? 2 squared times 2 cubed

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		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).

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: 2² × 2³ = 2^{2 + 3} = 2⁵

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

multiplying powers

Dividing Powers

When dividing the same number with exponents, subtract the exponents.

Example: 2⁵ ÷ 2³ = 2^{5 - 3} = 2²

2² = 2 × 2 = 4

dividing powers

Powers of a Power

When raising one exponent to another, multiply the exponents.

Example: (2²)³ = 2^{2 × 3} = 2⁶

2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64

finding a power of a power

Power of a Fraction

2 over 3 all to the n equals 2 to the n over 3 to the n

When 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 ⁄ 9

finding the power of fraction

Exponent Is Negative

2 to the minus n is equal to 1 divided by 2 to the n

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

negative exponents

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 = √(2³) = √(2 × 2 × 2) = √8 or

(√2)³ = √2 × √2 × √2 = √8

Lesson Slides

The 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.

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

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: 2 to the m plus 2 to the n, 2 to the m minus a to the n

Mathematics Monster has known some students who have got confused with other laws of exponents and have made up their own rules: a wrong law of exponents