Multiplying exponents

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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:

(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:

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