# Multiplying exponents

How to multiply exponents.

## Multiplying exponents with same base

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

a na m = a n+m

#### Example:

23 ⋅ 24 = 23+4 = 27 = 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 nb n = (a b) n

#### Example:

32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144

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

a nb m

#### Example:

32 ⋅ 43 = 9 ⋅ 64 = 576

## Multiplying negative exponents

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

a -na -m = a -(n+m) = 1 / a n+m

#### Example:

2-3 ⋅ 2-4 = 2-(3+4) = 2-7 = 1 / 27 = 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 -nb -n = (a b) -n

#### Example:

3-2 ⋅ 4-2 = (3⋅4)-2 = 12-2 = 1 / 122 = 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 -nb -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 = 45 / 35 = 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 = 0.83 = 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/mb n/m = (a b) n/m

#### Example:

23/2 ⋅ 33/2 = (2⋅3)3/2 = 63/2 = (63) = 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/mb k/j

#### Example:

2 3/2 ⋅ 24/3 = (23) ⋅ 3(24) = 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 = 56/2 = 53 = 125

## Multiplying variables with exponents

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

xnxm = xn+m

#### Example:

x2x3 = (x⋅x)(x⋅x⋅x) = x2+3 = x5

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