The Laws of Exponents in Algebra

The Laws of Exponents in Algebra

The 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.

a squared

In this example, a (called the base) is multiplied by itself 2 (the exponent) times.

a squared equals a times a

What if we see the a letter with an exponent multiplying that same letter with a different exponent?

a squared times a 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

a to the m times a to the n equals a to the m plus n

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

Example: a2 × a3 = a2 + 3 = a5

Read more about multiplying powers in algebra

Dividing Powers

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

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

Example: a5 ÷ a3 = a5 - 3 = a2

Read more about dividing powers in algbra

Powers of a Power

a to the m in brackets all to the n

When raising one exponent to another, multiply the exponents.

Example: (a2)3 = a2 × 3 = a6

Read more about finding a power of a power in algebra

Power of an Algebraic Fraction

a over b all to the n equals a to the n over b to the n

When raising a fraction to an exponent, raise both the numerator and denominator to the exponent.

Example: (a ⁄ b)2 = a2 ⁄ b2

Read more about finding the power of an algebraic fraction

Exponent Is Negative

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

A negative exponent means calculating the positive exponent and finding the reciprocal (i.e. find 1 over it).

Example: a−2 = 1 ⁄ a2

Read more about negative exponents in algebra

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: a½ = √a

Read more about how to find a fractional exponent in algebra

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: a32 = √(a3) = (√a)3

Read more about how to find a fractional exponent in algebra

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