Fractional exponents

In this article we will learn how to solve fractional exponents.

Simplifying fractional exponents

The base b raised to the power of n/m is equal to:

bn/m = (m√b)n = m√(bn)

For example:

34/2 = 2√(34) = 9

Simplifying fractions with exponents

Fractions with exponents:

(a / b)n = an / bn

For example:

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(6/3)3 = 63 / 33 = 216 / 27 = 8

Negative fractional exponents

The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power of n/m:

b-n/m = 1 / bn/m = 1 / (m√b)n

For example:

4-2/4 = 1/42/4 = 1/ 4√42 = 0.5

Fractions with negative exponents

The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power of n:

(a/b)-n = 1 / (a/b)n = 1 / (an/bn) = bn/an

For example:

(3/6)-2 = 1 / (6/3)2 = 1 / (62/32) = 32/62 = 9/36 = 0.25

Multiplying fractional exponents

Multiplying fractional exponents with same fractional exponent:

a n/m ⋅ b n/m = (a ⋅ b) n/m

For example:

44/2 ⋅ 54/2 = (4⋅5)4/2 = 204/2 = √(64) = √1296 = 36

Multiplying fractional exponents with same base:

a n/m ⋅ a k/j = a (n/m)+(k/j)

For example:

46/2 ⋅ 44/2 = 4(6/2)+(4/2) = 1024

Multiplying fractional exponents with different exponents and fractions:

a n/m ⋅ b k/j

For example:

34/2 ⋅ 28/4 = √(24) ⋅ 4√(38) = 4 ⋅ 9 = 36

Multiplying fractions with exponents

Multiplying fractions with exponents with same fraction base:

(a / b) n ⋅ (a / b) m = (a / b) n+m

For example:

(6/3)4 ⋅ (6/3)3 = (6/3)4+3 = (6/3)7 = 67 / 37 = 279936 / 2187 = 128

Multiplying fractions with exponents with same exponent:

(a / b) n ⋅ (c / d) n = ((a / b)⋅(c / d)) n

For example:

(6/3)2 ⋅ (3/6)2 = ((6/3)⋅(3/6))2 = (6/6)2 = 1

Multiplying fractions with exponents with different bases and exponents:

(a / b) n ⋅ (c / d) m

For example:

(3/1)2 ⋅ (2/2)3 = 9 / 1 = 9

Dividing fractional exponents

Dividing fractional exponents with same fractional exponent:

a n/m / b n/m = (a / b) n/m

For example:

64/2 / 34/2 = (6/3)4/2 = 24/2 = 4

Dividing fractional exponents with same base:

a n/m / a k/j = a (n/m)-(k/j)

For example:

24/3 / 26/5 = 2(4/3)-(6/5) = 2(13/100) = 100√213 = 1.09

Dividing fractional exponents with different exponents and fractions:

a n/m / b k/j

For example:

45/2 / 66/4 = √(25) / 4√(36) = 32 / 5.19 = 37.19

Dividing fractions with exponents

Dividing fractions with exponents with same fraction base:

(a / b)n / (a / b)m = (a / b)n-m

For example:

(6/2)4 / (6/2)3 = (6/2)4-3 = (6/2)1 = 6/2 = 3

Dividing fractions with exponents with same exponent:

(a / b)n / (c / d)n = ((a / b)/(c / d))n = ((a⋅d / b⋅c))n

For example:

(8/4)2 / (6/3)2 = ((8/4)/(6/3))2 = ((8⋅3)/(4⋅6))2 = (48/24)2 = 4

Dividing fractions with exponents with different bases and exponents:

(a / b) n / (c / d) m

For example:

(10/5)4 / (15/5)2 = 16 / 9 = 1.77

Adding fractional exponents

Adding fractional exponents is done by raising each exponent first and then adding:

an/m + bk/j

For example:

46/2 + 55/2 = √(46) + √(55) = √(4096) + √(3125) = 64 + 55.9 = 119.9

Adding same bases b and exponents n/m:

bn/m + bn/m = 2bn/m

For example:

53/4 + 53/4 = 2⋅53/4 = 2 ⋅ 4√(43) = 5.65

Subtracting fractional exponents

Subtracting fractional exponents is done by raising each exponent first and then subtracting:

an/m – bk/j

For example:

23/2 – 14/2 = √(23) – √(14) = √(8) – √(1) = 2.82 – 1 = 1.82

Subtracting same bases b and exponents n/m:

3bn/m – bn/m = 2bn/m

For example:

3⋅24/2 – 24/2 = 2⋅24/2 = 2 ⋅ √(24) = 8

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