# Dividing exponents

How to divide 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*)^{-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} / 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}

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} = √(*1.5*^{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} / 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}

## See also