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}*

For example:

*3 ^{4} / 3^{2} = 3^{4-2} = 3^{2} = 3⋅3 = 9*

### 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}*

For example:

*8 ^{4} / 4^{4} = (8/4)^{4} = 2^{4} = 2⋅2⋅2⋅2 = 16*

When the bases and the exponents are different we have to calculate each exponent and then divide:

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

For example:

*10 ^{2} / 5^{3} = 100 / 125 = 0.8*

### Dividing negative exponents

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

*a ^{-n} / a^{-m} = a^{-n-(-m)} = a^{m-n}*

For example:

*3 ^{-4} / 3^{-6} = 3^{6-4} = 3^{2} = 3⋅3 = 9*

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}*

For example:

*2 ^{-2} / 3^{-2} = (2/3)^{2} = 0.44*

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}*

For example:

*2 ^{-3} / 5^{-2} = 5^{2} / 2^{3} = 25 / 8 = 3.125*

### 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} = 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:

*(6/2) ^{2} / (4/2)^{2} = ((6/2)/(4/2))^{2} = ((6⋅2)/(2⋅4))^{3} = (12/8)^{3} = 3.375*

Dividing fractions with exponents with different bases and exponents:

*(a / b) ^{n} / (c / d)^{m}*

For example:

*(10/5) ^{4} / (9/3)^{3} = 16 / 27 = 0.592*

### 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} = √(2^{4}) = √16 = 4*

Dividing fractional exponents with same base:

*a ^{n/m} / a^{k/j} = a^{(n/m)-(k/j)}*

For example:

*4 ^{3/2} / 4^{4/3} = 4^{(3/2)-(4/3)} = 4^{(1/6)} = ^{6}√4 = 1.259*

Dividing fractional exponents with different exponents and fractions:

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

For example:

*3 ^{3/2} / 3^{4/3} = √(3^{3}) / ^{3}√(3^{4}) = 5.19 / 4.32 = 1.2*

### Dividing variables with exponents

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

*x ^{n} / x^{m} = x^{n-m}*

For example:

*x ^{3} / x^{2} = (x⋅x⋅x) / (x⋅x) = x^{3-2} = x^{1}*