Dividing exponents

How to divide exponents.

Dividing exponents with same base

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

aⁿ / a^m = a^n-m

For example:

3⁴ / 3² = 3^4-2 = 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:

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 / b^m

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) = a^m-n

For example:

3^-4 / 3^-6 = 3^6-4 = 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 = b^m / 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:

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

For example:

6^4/2 / 3^4/2 = (6/3)^4/2 = 2^4/2 = √(2⁴) = √16 = 4

Dividing fractional exponents with same base:

a^n/m / a^k/j = a^(n/m)-(k/j)

For example:

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

Dividing fractional exponents with different exponents and fractions:

a^n/m / b^k/j

For example:

3^3/2 / 3^4/3 = √(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 / x^m = x^n-m

For example:

x³ / x² = (x⋅x⋅x) / (x⋅x) = x^3-2 = x¹