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

**Contents**hide

- Simplifying fractional exponents
- Simplifying fractions with exponents
- Negative fractional exponents
- Fractions with negative exponents
- Multiplying fractional exponents
- Multiplying fractions with exponents
- Dividing fractional exponents
- Dividing fractions with exponents
- Adding fractional exponents
- Subtracting 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}√(b^{n})*

For example:

*3 ^{4/2} = ^{2}√(3^{4}) = 9*

### Simplifying fractions with exponents

Fractions with exponents:

*(a / b) ^{n} = a^{n} / b^{n}*

For example:

*(6/3) ^{3} = 6^{3} / 3^{3} = 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}√4^{2} = 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 / (a^{n}/b^{n}) = b^{n}/a^{n}*

For example:

*(3/6) ^{-2} = 1 / (6/3)^{2} = 1 / (6^{2}/3^{2}) = 3^{2}/6^{2} = 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} = √(6^{4}) = √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} = √(2^{4}) ⋅ ^{4}√(3^{8}) = 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} = 6^{7} / 3^{7} = 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}√2^{13} = 1.09*

Dividing fractional exponents with different exponents and fractions:

*a ^{n/m} / b ^{k/j}*

For example:

*4 ^{5/2} / 6^{6/4} = √(2^{5}) / ^{4}√(3^{6}) = 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} = √(4^{6}) + √(5^{5}) = √(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}√(4^{3}) = 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} = √(2^{3}) – √(1^{4}) = √(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 ⋅ √(2^{4}) = 8*