How to multiply exponents.

- Multiplying exponents with same base
- Multiplying exponents with different bases
- Multiplying negative exponents
- Multiplying fractions with exponents
- Multiplying fractional exponents
- Multiplying variables with exponents
- Multiplying square roots with exponents

## Multiplying exponents with same base

For exponents with the same base, we should add the exponents:

*a ^{ n}* ⋅

*a*=

^{ m}*a*

^{ n+m}#### Example:

2^{3} ⋅ 2^{4} = 2^{3+4} = 2^{7} = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

## Multiplying exponents with different bases

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}#### Example:

3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 12^{2} = 12⋅12 = 144

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

*a ^{ n}* ⋅

*b*

^{ m}#### Example:

3^{2} ⋅ 4^{3} = 9 ⋅ 64 = 576

## Multiplying negative exponents

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

*a ^{ -n}* ⋅

*a*=

^{ -m}*a*

^{ -(n+m}^{) }= 1 /

*a*

^{ n+m}#### Example:

2^{-3} ⋅ 2^{-4} = 2^{-(3+4)} = 2^{-7} = 1 / 2^{7} = 1 / (2⋅2⋅2⋅2⋅2⋅2⋅2) = 1 / 128 = 0.0078125

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}#### Example:

3^{-2} ⋅ 4^{-2} = (3⋅4)^{-2} = 12^{-2} = 1 / 12^{2} = 1 / (12⋅12) = 1 / 144 = 0.0069444

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

*a ^{ -n}* ⋅

*b*

^{ -m}#### Example:

3^{-2} ⋅ 4^{-3} = (1/9) ⋅ (1/64) = 1 / 576 = 0.0017361

## 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 = 0.5925

## 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} ⋅ 2^{4/3} = *√*(2^{3}) ⋅^{
3}*√*(2^{4})*
= *2.828 ⋅ 2.52* = *7.127

## Multiplying square roots with exponents

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

(√*a*)^{n} ⋅
(*√a*)^{m} = *
a*^{(n+m}^{)/2}

#### Example:

(√5)^{2} ⋅
(*√*5)^{4} = 5^{(2+4)/2} =
5^{6/2} = 5^{3} = 125

## Multiplying variables with exponents

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

*x ^{n}* ⋅

*x*=

^{m}*x*

^{n+m}#### Example:

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

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