Dividing exponents

How to divide exponents.

• Dividing exponents with same base
• Dividing exponents with different bases
• Dividing negative exponents
• Dividing fractions with exponents
• Dividing fractional exponents
• Dividing variables with exponents
• Dividing square roots with exponents

Dividing exponents with same base

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

a ⁿ / a ^m = a ^n-m

Example:

2⁶ / 2³ = 2^6-3 = 2³ = 2⋅2⋅2 = 8

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:

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

Dividing negative exponents

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

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

Example:

2^-3 / 2^-5 = 2^5-3 = 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 = b^m / aⁿ

Example:

3^-2 / 4^-3 = 4³ / 3² = 64 / 9 = 7.111

fractions with exponents

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

Example:

(4/3)³ / (1/2)² = 2.37 / 0.25 = 9.481

Dividing fractional exponents

Dividing fractional exponents with same fractional exponent:

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

Example:

3^3/2 / 2^3/2 = (3/2)^3/2 = 1.5^3/2 = √(1.5³) = √3.375 = 1.837

Dividing fractional exponents with same base:

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

Example:

2^3/2 / 2^4/3 = 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

Example:

2^3/2 / 2^4/3 = √(2³) / ³√(2⁴) = 2.828 / 2.52 = 1.1222

Dividing variables with exponents

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

xⁿ / x^m = x^n-m

Example:

x⁵ / x³ = (x⋅x⋅x⋅x⋅x) / (x⋅x⋅x) = x^5-3 = x²