# 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 / *b*^{n}

## 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} = 1/2^{3} = 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 / (*a*^{n}/*b*^{n})
= *b*^{n}/*a*^{n}

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 / (2^{2}/3^{2})
= 3^{2}/2^{2 }= 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 / 2^{7} = 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 / 12^{2} = 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:

*a*^{ n} / *a*^{ m} = *a*^{ n-m}

Example:

2^{6} / 2^{3} = 2^{6-3} = 2^{3} = 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:

*a*^{ n} / *b*^{ n} = (*a
/ b*)^{ n}

Example:

6^{3} / 2^{3} = (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:

*a*^{ n} / *b*^{ m}

Example:

6^{2} / 3^{3} = 36 / 27 = 1.333

## See also