How to divide 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 different 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
For exponents with the same base, we can subtract the exponents:
a-n / a-m = a-n-(-m) = am-n
Example:
2-3 / 2-5 = 25-3 = 22 = 2⋅2 = 4
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
Example:
3-2 / 4-2 = (4/3)2 = 1.7778
When the bases and the exponents are different we have to calculate each exponent and then divide:
a-n / b-m = bm / an
Example:
3-2 / 4-3 = 43 / 32 = 64 / 9 = 7.111
Dividing fractions with exponents with same fraction base:
(a / b)n / (a / b)m = (a / b)n-m
Example:
(4/3)3 / (4/3)2 = (4/3)3-2 = (4/3)1 = 4/3 = 1.333
Dividing fractions with exponents with same exponent:
(a / b)n / (c / d)n = ((a / b)/(c / d))n = ((a⋅d / b⋅c))n
Example:
(4/3)3 / (3/5)3 = ((4/3)/(3/5))3 = ((4⋅5)/(3⋅3))3 = (20/9)3 = 10.97
Dividing fractions with exponents with different bases and exponents:
(a / b) n / (c / d) m
(4/3)3 / (1/2)2 = 2.37 / 0.25 = 9.481
Dividing fractional exponents with same fractional exponent:
a n/m / b n/m = (a / b) n/m
Example:
33/2 / 23/2 = (3/2)3/2 = 1.53/2 = √(1.53) = √3.375 = 1.837
Dividing fractional exponents with same base:
a n/m / a k/j = a (n/m)-(k/j)
Example:
23/2 / 24/3 = 2(3/2)-(4/3) = 2(1/6) = 6√2 = 1.122
Dividing fractional exponents with different exponents and fractions:
a n/m / b k/j
23/2 / 24/3 = √(23) / 3√(24) = 2.828 / 2.52 = 1.1222
For exponents with the same base, we can subtract the exponents:
xn / xm = xn-m
Example:
x5 / x3 = (x⋅x⋅x⋅x⋅x) / (x⋅x⋅x) = x5-3 = x2