## 1. 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}Example:

The base 2 raised to the power of 3/2 is equal to 1 divided by the base 2 raised to the power of 3:

2^{3/2} = ^{2}*√*(2^{3}) = 2.828

## 2. Simplifying fractions with exponents:

Fractions with exponents:

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

*a*/

^{n}*b*

^{n}Example:

(4/3)^{3} = 4^{3 }/ 3^{3} = 64 / 27 = 2.37

## 3. 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}

Example:

The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:

2^{-1/2} = 1/2^{1/2} = 1/*√*2 = 0.7071

## 4. 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}

Example:

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

(2/3)^{-2} = 1 / (2/3)^{2} = 1 / (2^{2}/3^{2}) = 3^{2}/2^{2 }= 9/4 = 2.25

## 5. Multiplying fractional exponents:

Multiplying fractional exponents with same fractional exponent:

*a ^{ n/m}* ⋅

*b*= (

^{ n/m}*a*⋅

*b*)

^{ n/m}Example:

2^{3/2} ⋅ 3^{3/2} = (2⋅3)^{3/2}* = *6^{3/2}* =**√*(6^{3}) = *√*216 = 14.7

Multiplying 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)}* = *7.127

Multiplying fractional exponents with different exponents and fractions:

*a ^{ n/m}* ⋅

*b*

^{ k/j}Example:

2^{3/2} ⋅ 3^{4/3} = *√*(2^{3}) ⋅^{ 3}*√*(3^{4})* =*2.828 ⋅ 4.327* = *12.237

## 6. Multiplying fractions with exponents:

Multiplying 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)^{5} = 4^{5} / 3^{5} = 4.214

Multiplying fractions with exponents with same exponent:

(*a / b*)* ^{ n}* ⋅ (

*c / d*)

*= ((*

^{ n}*a / b*)⋅(

*c / d*))

^{ n}Example:

(4/3)^{3} ⋅ (3/5)^{3} = ((4/3)⋅(3/5))^{3}= (4/5)^{3} = 0.8^{3} = 0.8⋅0.8⋅0.8 = 0.512

Multiplying 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

## 7. 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} =√*(1.5

*) =*

^{3}*√*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} / 3^{4/3} = *√*(2^{3}) /^{ 3}*√*(3^{4})* =*2.828 / 4.327* = *0.654

## 8. Dividing 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

## 9. Adding fractional exponents:

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

*a ^{n/m}* +

*b*

^{k/j}Example:

3^{3/2} + 2^{5/2} = √(3^{3}) + √(2^{5}) = √(27) + √(32) = 5.196 + 5.657 = 10.853

Adding same bases b and exponents n/m:

*b ^{n/m}* +

*b*= 2

^{n/m}*b*

^{n/m}Example:

4^{2/3} + 4^{2/3} = 2⋅4^{2/3} = 2 ⋅^{3}√(4^{2}) = 5.04

## 10. Subtracting fractional exponents:

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

*a ^{n/m}* -

*b*

^{k/j}Example:

3^{3/2} - 2^{5/2} = √(3^{3}) - √(2^{5}) = √(27) - √(32) = 5.196 - 5.657 = -0.488

Subtracting same bases b and exponents n/m:

3*b ^{n/m}* -

*b*= 2

^{n/m}*b*

^{n/m}Example:

3⋅4^{2/3} - 4^{2/3} = 2⋅4^{2/3} = 2 ⋅^{3}√(4^{2}) = 5.04