Simplify the third root of 4 over the fifth root of 8
4^(1/3) = 2^(2/3)
8^(1/5) = 2^(3/5)
so, we have 2^(2/3) / 2^(3/5) = 2^(1/15) = fifteenthroot(2)
To simplify the expression, we can start by simplifying the individual radicals.
The third root of 4 can be written as 4^(1/3), and the fifth root of 8 can be written as 8^(1/5).
Now, let's simplify each of these expressions:
4^(1/3) = (2^2)^(1/3) = 2^(2/3)
8^(1/5) = (2^3)^(1/5) = 2^(3/5)
Now, we can rewrite the expression using the simplified radicals:
(2^(2/3))/(2^(3/5))
To divide two terms with the same base (in this case, 2), we subtract the exponents:
2^(2/3 - 3/5)
To subtract the exponents, we need a common denominator. The least common multiple of 3 and 5 is 15, so we rewrite the exponents with the common denominator:
2^(10/15 - 9/15)
Now, we can subtract the exponents:
2^(1/15)
Thus, the simplified form of the given expression is the 15th root of 2.