Simplify expression. Answers should only have positive exponents. Assume all variables are positive.
(4x^2 y^-1/3)^3/14
so would it be x^2(3/14) and y^-1/3(3/14)?
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.
To simplify the expression (4x^2y^(-1/3))^3/14 and rewrite it with positive exponents, we can follow these steps:
Step 1: Distribute the exponent outside the parentheses to each term inside the parentheses.
(4x^2y^(-1/3))^3/14 becomes (4^(3/14))(x^(2*(3/14)))(y^(-1/3*(3/14)))
Step 2: Simplify the exponents.
(4^(3/14)) is already in its simplest form.
(x^(2*(3/14))) is equivalent to x^(6/14), which simplifies to x^(3/7).
(y^(-1/3*(3/14))) is equivalent to y^(-3/42), which simplifies to y^(-1/14).
Step 3: Combine the simplified terms.
The simplified expression is (4^(3/14))(x^(3/7))(y^(-1/14)).
So, the correct answer is (4^(3/14))(x^(3/7))(y^(-1/14)).