## 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:

2^{6} / 2^{3} = 2^{6-3} = 2^{3} = 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:

6^{3} / 2^{3} = (6/2)^{3} = 3^{3} = 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:

6^{2} / 3^{3} = 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} = 2^{2} = 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*)

*1 / (*

^{-n}=*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} / a^{n}*

Example:

3^{-2} / 4^{-3} = 4^{3} / 3^{2} = 64 / 9 = 7.111

## 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}(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}= √*1.5*) =

^{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}2^{3/2} / 2^{4/3} = *√*(2^{3}) /
^{ 3}*√*(2^{4})* = *2.828 / 2.52* = * 1.1222

## Dividing variables with exponents

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

*x ^{n}* /

*x*=

^{m}*x*

^{n-m}Example:

*x*^{5} / *x*^{3}* = * (
*x⋅x⋅x⋅x⋅x*)* / *(*x⋅x⋅x*)* = x*^{5-3}* = x*^{2}