Fractional exponents

How to solve fractional exponents.

Simplifying fractional exponents

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

bn/m = (mb)n = m(bn)

Example:

The base 2 raised to the power of 3/2 is equal to 1 divided by the base 2 raised to the power of 3:

23/2 = 2(23) = 2.828

Simplifying fractions with exponents

Fractions with exponents:

(a / b)n = an / bn

Example:

(4/3)3 = 43 / 33 = 64 / 27 = 2.37

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 / (mb)n

Example:

The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:

2-1/2 = 1/21/2 = 1/2 = 0.7071

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

Example:

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

(2/3)-2 = 1 / (2/3)2 = 1 / (22/32) = 32/22 = 9/4 = 2.25

Multiplying fractional exponents

Multiplying fractional exponents with same fractional exponent:

a n/mb 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/ma k/j = a (n/m)+(k/j)

Example:

23/2 ⋅ 24/3 = 2(3/2)+(4/3) = 7.127

Multiplying fractional exponents with different exponents and fractions:

a n/mb k/j

Example:

23/2 ⋅ 34/3 = (23) ⋅ 3(34) = 2.828 ⋅ 4.327 = 12.237

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 = 9.481

Dividing fractional exponents

Dividing fractional exponents with same fractional exponent:

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

Example:

33/2 / 23/2 = (3/2)3/2 = 1.53/2 = (1.53) = 3.375 = 1.837

Dividing 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(1/6) = 62 = 1.122

Dividing fractional exponents with different exponents and fractions:

a n/m / b k/j

Example:

23/2 / 34/3 = (23) / 3(34) = 2.828 / 4.327 = 0.654

Dividing fractions with exponents

Dividing 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)1 = 4/3 = 1.333

Dividing fractions with exponents with same exponent:

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

Example:

(4/3)3 / (3/5)3 = ((4/3)/(3/5))3 = ((4⋅5)/(3⋅3))3 = (20/9)3 = 10.97

Dividing 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 = 9.481

Adding fractional exponents

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

an/m + bk/j

Example:

33/2 + 25/2 = √(33) + √(25) = √(27) + √(32) = 5.196 + 5.657 = 10.853

Adding same bases b and exponents n/m:

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

Example:

42/3 + 42/3 = 2⋅42/3 = 2 ⋅ 3√(42) = 5.04

Subtracting fractional exponents

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

an/m - bk/j

Example:

33/2 - 25/2 = √(33) - √(25) = √(27) - √(32) = 5.196 - 5.657 = -0.488

Subtracting same bases b and exponents n/m:

3bn/m - bn/m = 2bn/m

Example:

3⋅42/3 - 42/3 = 2⋅42/3 = 2 ⋅ 3√(42) = 5.04


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