Answer:
1/64
Step-by-step explanation:
Simplify the following:
1/(4^(11/3)×4^(-2/3))
Hint: | Simplify radicals.
1/4^(2/3) = 1/(4^2)^(1/3) = 1/(2^4)^(1/3) = (1/2^(1/3))/2:
1/(4^(11/3)×(1/2^(1/3))/2)
Hint: | Multiply numerator and denominator of 1/(2 2^(1/3)) by 2^(2/3).
Rationalize the denominator. 1/(2 2^(1/3)) = 1/(2 2^(1/3))×(2^(2/3))/(2^(2/3)) = (2^(2/3))/(2×2):
1/(4^(11/3)×(2^(2/3))/(2×2))
Hint: | Multiply 2 and 2 together.
2×2 = 4:
1/(4^(11/3)×(2^(2/3))/4)
Hint: | Separate the exponent of 4^(-11/3) into integer and fractional parts.
4^(-11/3) = 4^(1/3 - 12/3) = 4^(-12/3)×4^(1/3):
(4^(-12/3) 4^(1/3))/((2^(2/3))/4)
Hint: | Divide 12 by 3.
12/3 = (3 (-4))/3 = 4:
(4^(-4) 4^(1/3))/((2^(2/3))/4)
Hint: | Compute 4^4 by repeated squaring.
4^4 = (4^2)^2:
(4^(1/3))/((2^(2/3))/4) 1/(4^2)^2
Hint: | Evaluate 4^2.
4^2 = 16:
(4^(1/3))/(16^2×(2^(2/3))/4)
Hint: | Evaluate 16^2.
| 1 | 6
× | 1 | 6
| 9 | 6
1 | 6 | 0
2 | 5 | 6:
(4^(1/3))/(256×(2^(2/3))/4)
Hint: | Simplify radicals.
4^(1/3) = (2^2)^(1/3):
1/(256×(2^(2/3))/4) 2^(2/3)
Hint: | Write (2^(2/3))/(256×(2^(2/3))/4) as a single fraction.
Multiply the numerator by the reciprocal of the denominator, (2^(2/3))/(256×(2^(2/3))/4) = (2^(2/3))/256×4/2^(2/3):
(2^(2/3)×4)/(256×2^(2/3))
Hint: | Cancel common terms in the numerator and denominator of (2^(2/3)×4)/(256×2^(2/3)).
(2^(2/3)×4)/(256×2^(2/3)) = (2^(2/3))/(2^(2/3))×4/256 = 4/256:
4/256
Hint: | Reduce 4/256 to lowest terms. Start by finding the GCD of 4 and 256.
The gcd of 4 and 256 is 4, so 4/256 = (4×1)/(4×64) = 4/4×1/64 = 1/64:
Answer: 1/64
