How to calculate negative exponents.
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
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
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 / (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 = 1/21/2 = 1/√2 = 0.7071
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
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
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
