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 = 2·2·2·2·2·2·2 = 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 = (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 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 / (2·2·2·2·2·2·2) = 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 = (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 -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 = 0.8·0.8·0.8 = 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:
23/2 · 3
3/2 = (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:
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 = 5
6/2 = 53 = 125
Multiplying variables with exponents
For exponents with the same base, we can add the exponents:
xn · xm = xn+m
Example:
x2 · x3 = (
x·x) · (x·x·x) = x2+3 = x5