Multiplying exponents

How to multiply 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:

23 24 = 23+4 = 27 = 2222222 = 128

Multiplying exponents with different bases

When the bases are diffenrent 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:

32 42 = (34)2 = 122 = 1212 = 144

 

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

a n b m

Example:

32 43 = 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 / 27 = 1 / (2222222) = 1 / 128 = 0.0078125

 

When the bases are diffenrent 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 = (34)-2 = 12-2 = 1 / 122 = 1 / (1212) = 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 = 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 = 0.80.80.8 = 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:

23/2 33/2 = (23)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:

23/2 24/3 = 2(3/2)+(4/3) = 2.8333

 

Multiplying fractional exponents with different exponents and fractions:

a n/m b k/j

Example:

23/2 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:

xn xm = xn+m

Example:

x2 x3 = (xx) (xxx) = x2+3 = x5

 


See also

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