Fractional exponents

In this article we will learn how to solve fractional exponents.

Simplifying fractional exponents

The base b raised to the power of n/m is equal to:

b^n/m = (^m√b)ⁿ = ^m√(bⁿ)

For example:

3^4/2 = ²√(3⁴) = 9

Simplifying fractions with exponents

Fractions with exponents:

(a / b)ⁿ = aⁿ / bⁿ

For example:

(6/3)³ = 6³ / 3³ = 216 / 27 = 8

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 / b^n/m = 1 / (^m√b)ⁿ

For example:

4^-2/4 = 1/4^2/4 = 1/ ⁴√4² = 0.5

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:

(3/6)^-2 = 1 / (6/3)² = 1 / (6²/3²) = 3²/6² = 9/36 = 0.25

Multiplying fractional exponents

Multiplying fractional exponents with same fractional exponent:

a ^n/m ⋅ b ^n/m = (a ⋅ b) ^n/m

For example:

4^4/2 ⋅ 5^4/2 = (4⋅5)^4/2 = 20^4/2 = √(6⁴) = √1296 = 36

Multiplying fractional exponents with same base:

a ^n/m ⋅ a ^k/j = a ^(n/m)+(k/j)

For example:

4^6/2 ⋅ 4^4/2 = 4^(6/2)+(4/2) = 1024

Multiplying fractional exponents with different exponents and fractions:

a ^n/m ⋅ b ^k/j

For example:

3^4/2 ⋅ 2^8/4 = √(2⁴) ⋅ ⁴√(3⁸) = 4 ⋅ 9 = 36

Multiplying fractions with exponents

Multiplying fractions with exponents with same fraction base:

(a / b) ⁿ ⋅ (a / b) ^m = (a / b) ^n+m

For example:

(6/3)⁴ ⋅ (6/3)³ = (6/3)⁴⁺³ = (6/3)⁷ = 6⁷ / 3⁷ = 279936 / 2187 = 128

Multiplying fractions with exponents with same exponent:

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

For example:

(6/3)² ⋅ (3/6)² = ((6/3)⋅(3/6))² = (6/6)² = 1

Multiplying fractions with exponents with different bases and exponents:

(a / b) ⁿ ⋅ (c / d) ^m

For example:

(3/1)² ⋅ (2/2)³ = 9 / 1 = 9

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 = 4

Dividing fractional exponents with same base:

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

For example:

2^4/3 / 2^6/5 = 2^(4/3)-(6/5) = 2^(13/100) = ¹⁰⁰√2¹³ = 1.09

Dividing fractional exponents with different exponents and fractions:

a ^n/m / b ^k/j

For example:

4^5/2 / 6^6/4 = √(2⁵) / ⁴√(3⁶) = 32 / 5.19 = 37.19

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 = 6/2 = 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:

(8/4)² / (6/3)² = ((8/4)/(6/3))² = ((8⋅3)/(4⋅6))² = (48/24)² = 4

Dividing fractions with exponents with different bases and exponents:

(a / b) ⁿ / (c / d) ^m

For example:

(10/5)⁴ / (15/5)² = 16 / 9 = 1.77

Adding fractional exponents

Adding fractional exponents is done by raising each exponent first and then adding:

a^n/m + b^k/j

For example:

4^6/2 + 5^5/2 = √(4⁶) + √(5⁵) = √(4096) + √(3125) = 64 + 55.9 = 119.9

Adding same bases b and exponents n/m:

b^n/m + b^n/m = 2b^n/m

For example:

5^3/4 + 5^3/4 = 2⋅5^3/4 = 2 ⋅ ⁴√(4³) = 5.65

Subtracting fractional exponents

Subtracting fractional exponents is done by raising each exponent first and then subtracting:

a^n/m – b^k/j

For example:

2^3/2 – 1^4/2 = √(2³) – √(1⁴) = √(8) – √(1) = 2.82 – 1 = 1.82

Subtracting same bases b and exponents n/m:

3b^n/m – b^n/m = 2b^n/m

For example:

3⋅2^4/2 – 2^4/2 = 2⋅2^4/2 = 2 ⋅ √(2⁴) = 8