# Negative exponents

How to calculate negative exponents.

- Negative exponents rule
- Negative exponent example
- Negative fractional exponents
- Fractions with negative exponents
- Multiplying negative exponents
- Dividing 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