How to multiply exponents.
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
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
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 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
(4/3)3 ⋅ (1/2)2 = 2.37 ⋅ 0.25 = 0.5925
Multiplying fractional exponents with same fractional exponent:
a n/m ⋅ b 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/m ⋅ b k/j
2 3/2 ⋅ 24/3 = √(23) ⋅ 3√(24) = 2.828 ⋅ 2.52 = 7.127
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
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