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


Currently, we have around 929 calculators and conversion tables to help you "do the math" quickly in areas such as:

And we are still developing more. Our goal is to become the one-stop, go-to site for people who need to make quick calculations or who need to find quick answer for basic conversions.

Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. We coded and developed each calculator individually and put each one through strict, comprehensive testing. However, please inform us if you notice even the slightest error – your input is extremely valuable to us. While most calculators on Justfreetools.com are designed to be universally applicable for worldwide usage, some are for specific countries only.


This website uses cookies to improve your experience, analyze traffic and display ads. Learn more