How to solve a problem with the number race to the power of some fraction
Simplifying fractional exponents The base b raised to the power of n/m is equal to:
bn/m = (m√b)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 /(m√b)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/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:
23/2 ⋅ 24/3 = 2(3/2)+(4/3) = 7.127
Multiplying 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 = 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)= 6√2 = 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
The base b raised to the power of n/m is equal to:
bn/m = (m√b)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 exponentsFractions with exponents:
(a / b)n = an / bn
Example:
(4/3)3 = 43 / 33 = 64 / 27 = 2.37
Negative fractional exponentsThe 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
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 exponentsThe 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 exponentsMultiplying 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:
23/2 ⋅ 24/3 = 2(3/2)+(4/3) = 7.127
Multiplying 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 = 12.237
Multiplying fractions with exponentsMultiplying 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 exponentsDividing 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)= 6√2 = 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 exponentsDividing 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 exponentsAdding 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 exponentsSubtracting 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
