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:

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 1 to the n equals 1 14 = 1 × 1 × 1 × 1 = 1
Exponent of 0 a to the 0 equals 1 Any base with an exponent of 0 is 1.
Exponent of 1 a to the 1 equals a Any base with an exponent of 1 is equal to the base.
Exponent of −1 a to the minus 1 equals 1 divided by a 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

2 to the m times 2 to the n equals 2 to the m plus n

When multiplying the same number with exponents, add the exponents.

Example: 22 × 23 = 22 + 3 = 25

25 = 2 × 2 × 2 × 2 × 2 = 32

Read more about multiplying powers

Dividing Powers

2 to the m divided by 2 to the n equals 2 to the m minus n

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

Example: 25 ÷ 23 = 25 - 3 = 22

22 = 2 × 2 = 4

Read more about dividing powers

Powers of a Power

2 to the m in brackets all to the n

When raising one exponent to another, multiply the exponents.

Example: (22)3 = 22 × 3 = 26

26 = 2 × 2 × 2 × 2 × 2 × 2 = 64

Read more about 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

Read more about 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

Read more about negative exponents

Exponent Is a Fraction (Numerator is 1)

a to the 1 over n

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)

a to the m over n

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: 232 = √(23) = √(2 × 2 × 2) = √8 or

(√2)3 = √2 × √2 × √2 = √8

Slider

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.

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See Also

What is an exponent? Finding a negative exponent Multiplying powers Dividing powers Finding a power of a fraction Finding a power of a power