Negative exponents

How to calculate negative exponents.

Negative exponents rule

The base b raised to the power of minus n is equal to 1 divided by the base b raised to the power of n:

b^-n = 1 / bⁿ

Negative exponent example

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

2^-3 = 1/2³ = 1/(2⋅2⋅2) = 1/8 = 0.125

Negative fractional exponents

The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power of n/m:

b^-n/m = 1 / b^n/m = 1 / ( ^m√b)ⁿ

The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:

2^-1/2 = 1/2^1/2 = 1/√2 = 0.7071

Fractions with negative exponents

The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power of n:

(a/b)^-n = 1 / ( a/b)ⁿ = 1 / (aⁿ/bⁿ) = bⁿ/aⁿ

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

(2/3)^-2 = 1 / (2/3)² = 1 / (2²/3²) = 3²/2²= 9/4 = 2.25

Multiplying negative exponents

For exponents with the same base, we can add the exponents:

a^-n ⋅ a^-m = a^-(n+m⁾= 1 / a^n+m

Example:

2^-3 ⋅ 2^-4 = 2^-(3+4) = 2^-7 = 1 / 2⁷ = 1 / (2⋅2⋅2⋅2⋅2⋅2⋅2) = 1 / 128 = 0.0078125

When the bases are diffenrent and the exponents of a and b are the same, we can multiply a and b first:

a^-n ⋅ b^-n = (a ⋅ b)^-n

Example:

3^-2 ⋅ 4^-2 = (3⋅4)^-2 = 12^-2 = 1 / 12² = 1 / (12⋅12) = 1 / 144 = 0.0069444

When the bases and the exponents are different we have to calculate each exponent and then multiply:

a^-n ⋅ b^-m

Example:

3^-2 ⋅ 4^-3 = (1/9) ⋅ (1/64) = 1 / 576 = 0.0017361

Dividing negative exponents

For exponents with the same base, we should subtract the exponents:

aⁿ / a^m = a^n-m

Example:

2⁶ / 2³ = 2^6-3 = 2³ = 2⋅2⋅2 = 8

When the bases are diffenrent and the exponents of a and b are the same, we can divide a and b first:

aⁿ / bⁿ = (a/ b)ⁿ

Example:

6³ / 2³ = (6/2)³ = 3 ³ = 3⋅3⋅3 = 27

When the bases and the exponents are different we have to calculate each exponent and then divide:

aⁿ / b^m

Example:

6² / 3³ = 36 / 27 = 1.333