Negative exponents

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 / bn

For example:

3-3 = 1/33 = 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 / bn/m = 1 / (m√b)n

For example:

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3-1/2 = 1/31/2 = 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 / (an/bn) = bn/an

For example:

(3/4)-2 = 1 / (3/4)2 = 1 / (32/42) = 42/3= 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 / 45 = 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 / 63 = 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:

35 / 32 = 35-2 = 33 = 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:

34 / 24 = (3/2)4 = 1.54 = 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:

82 / 44 = 64 / 256 = 0.25

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