To simplify the expression (4x^2 y^-1/3)^(3/14), we can rewrite it using the power of a product rule, which states that (ab)^c = a^c * b^c.
So, applying the rule, we have:
(4x^2 y^-1/3)^(3/14) = (4^3/14) * (x^2)^(3/14) * (y^-1/3)^(3/14)
Now let's simplify each term separately:
(4^3/14) = 4^(3/14) ≈ 1.371
Using a calculator, we find that 4^(3/14) is approximately equal to 1.371.
(x^2)^(3/14) = x^(2 * 3/14) = x^(3/7)
We multiply the exponents, 2 and 3/14, to simplify the expression.
(y^-1/3)^(3/14) = y^(-1/3 * 3/14) = y^(-1/7)
We multiply the exponents, -1/3 and 3/14, to simplify the expression.
Therefore, simplifying the original expression:
(4x^2 y^-1/3)^(3/14) ≈ 1.371 * x^(3/7) * y^(-1/7)
Note that we have represented the answer in decimal form for the coefficient (1.371), but the exponents are still in fractional form.