Dividing exponents

How to divide exponents.

Dividing exponents with same base

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

an / am = an-m

For example:

34 / 32 = 34-2 = 32 = 3⋅3 = 9

Dividing exponents with different bases

When the bases are different and the exponents of a and b are the same, we can divide a and b first:

an / bn = (a / b)n

For example:

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84 / 44 = (8/4)4 = 24 = 2⋅2⋅2⋅2 = 16

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

an / bm

For example:

102 / 53 = 100 / 125 = 0.8

Dividing negative exponents

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

a-n / a-m = a-n-(-m) = am-n

For example:

3-4 / 3-6 = 36-4 = 32 = 3⋅3 = 9

When the bases are different and the exponents of a and b are the same, we can multiply a and b first:

a-n / b-n = (a/b)-n = 1 / (a/b)n = (b/a)n

For example:

2-2 / 3-2 = (2/3)2 = 0.44

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

a-n / b-m = bm / an

For example:

2-3 / 5-2 = 52 / 23 = 25 / 8 = 3.125

Dividing fractions with exponents

Dividing fractions with exponents with same fraction base:

(a / b)n / (a / b)m = (a / b)n-m

For example:

(6/2)4 / (6/2)3 = (6/2)4-3 = (6/2)1 = 3

Dividing fractions with exponents with same exponent:

(a / b)n / (c / d)n = ((a / b)/(c / d))n = ((a⋅d / b⋅c))n

For example:

(6/2)2 / (4/2)2 = ((6/2)/(4/2))2 = ((6⋅2)/(2⋅4))3 = (12/8)3 = 3.375

Dividing fractions with exponents with different bases and exponents:

(a / b)n / (c / d)m

For example:

(10/5)4 / (9/3)3 = 16 / 27 = 0.592

Dividing fractional exponents

Dividing fractional exponents with same fractional exponent:

an/m / bn/m = (a / b)n/m

For example:

64/2 / 34/2 = (6/3)4/2 = 24/2 = √(24) = √16 = 4

Dividing fractional exponents with same base:

an/m / ak/j = a(n/m)-(k/j)

For example:

43/2 / 44/3 = 4(3/2)-(4/3) = 4(1/6) = 6√4 = 1.259

Dividing fractional exponents with different exponents and fractions:

an/m / bk/j

For example:

33/2 / 34/3 = √(33) / 3√(34) = 5.19 / 4.32 = 1.2

Dividing variables with exponents

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

xn / xm = xn-m

For example:

x3 / x2 = (x⋅x⋅x) / (x⋅x) = x3-2 = x1

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