Negative Components

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 / bn/m = 1 /(mb)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 = 1/21/2 = 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:

a n / a m = a n-m

Example:

26 / 23 = 26-3 = 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:

a n / b n = (a / b) n

Example:

63 / 23 = (6/2)3 = 33 = 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:

62 / 33 = 36 / 27 = 1.333