In this article we will learn 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}*

For example:

*3 ^{-3} = 1/3^{3} = 1/(3⋅3⋅3) = 1/27 = 0.037*

### 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}*

For example:

*3 ^{-1/2} = 1/3^{1/2} = 1/√3 = 0.57*

### 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}*

For example:

*(3/4) ^{-2} = 1 / (3/4)^{2} = 1 / (3^{2}/4^{2}) = 4^{2}/3^{2 }= 16/9 = 1.77*

### 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}*

For example:

*4 ^{-2} ⋅ 4^{-3} = 4^{-(2+3)} = 4^{-5} = 1 / 4^{5} = 1 / (4⋅4⋅4⋅4⋅4) = 1 / 1024 = 0.00097*

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}*

For example:

*2 ^{-3} ⋅ 3^{-3} = (2⋅3)^{-3} = 6^{-3} = 1 / 6^{3} = 1 / (6⋅6⋅6) = 1 / 216 = 0.00462*

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

*a ^{-n} ⋅ b ^{-m}*

For example:

*5 ^{-2} ⋅ 6^{-3} = (1/25) ⋅ (1/216) = 1 / 576 = 0.00000803*

### Dividing negative exponents

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

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

For example:

*3 ^{5} / 3^{2} = 3^{5-2} = 3^{3} = 3⋅3⋅3 = 27*

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}*

For example:

*3 ^{4} / 2^{4} = (3/2)^{4} = 1.5^{4} = 1.5⋅1.5⋅1.5 = 3.375*

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

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

For example:

*8 ^{2} / 4^{4} = 64 / 256 = 0.25*