In this article we will learn 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
For example:
32 ⋅ 33 = 32+3 = 35 = 3⋅3⋅3⋅3⋅3 = 243
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
For example:
23 ⋅ 33 = (2⋅3)3 = 63 = 6⋅ 6⋅ 6 = 216
When the bases and the exponents are different we have to calculate each exponent and then multiply:
a n ⋅ b m
For example:
13 ⋅ 22 = 1 ⋅ 4 = 4
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
For example:
1-2 ⋅ 1-3 = 1-(2+3) = 1-5 = 1 / 15 = 1 / (1⋅1⋅1⋅1⋅1) = 1 / 1 = 1
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
For example:
2-3 ⋅ 3-3 = (2⋅3)-3 = 6-3 = 1 / 63 = 1 / 216 = 0.00462
When the bases and the exponents are different we have to calculate each exponent and then multiply:
a -n ⋅ b -m
For example:
2-2 ⋅ 1-3 = (1/4) ⋅ (1/1) = 1 / 4 = 0.25
Multiplying fractions with exponents
Multiplying fractions with exponents with same fraction base:
(a / b) n ⋅ (a / b) m = (a / b) n+m
For example:
(2/4)3 ⋅ (2/4)2 = (2/4)3+2 = (2/4)5 = 25 / 45 = 32 + 1024 = 1056
Multiplying fractions with exponents with same exponent:
(a / b) n⋅ (c / d) n = ((a / b)⋅(c / d)) n
For example:
(4/2)3 ⋅ (6/3)3 = ((4/2)⋅(6/3))3 = 43 = 64
Multiplying fractions with exponents with different bases and exponents:
(a / b) n ⋅ (c / d) m
For example:
(2/4)3 ⋅ (4/2)2 = 0.125 ⋅ 4 = 0.5
Multiplying fractional exponents
Multiplying fractional exponents with same fractional exponent:
a n/m ⋅ b n/m = (a ⋅ b) n/m
For example:
43/2 ⋅ 23/2 = (4⋅2)3/2 = 83/2 = √(83) = √216 = 22.6
Multiplying fractional exponents with same base:
a (n/m) ⋅ a (k/j) = a [(n/m)+(k/j)]
For example:
4(6/3) ⋅ 4(4/2) = 4[(6/3)+(4/2)] = 256
Multiplying fractional exponents with different exponents and fractions:
a n/m ⋅ b k/j
For example:
4 6/2 ⋅ 34/3 = √(46) ⋅ 3√(34) = 64 ⋅ 4.32 = 68.32
Multiplying square roots with exponents
For exponents with the same base, we can add the exponents:
(√a)n ⋅ (√a)m = a(n+m)/2
For example:
(√4)2 ⋅ (√4)4 = 4(2+4)/2 = 46/2 = 43 = 64
Multiplying variables with exponents
For exponents with the same base, we can add the exponents:
xn ⋅ xm = xn+m
For example:
x3 ⋅ x4 = (x⋅x⋅x) ⋅ (x⋅x⋅x⋅x) = x3+4 = x7