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, you need to apply the exponent rules and simplify each part of the expression separately.
Let's start with (4x^2 y^-1/3). Raised to the power of 3/14, each term inside the parentheses will be raised to that exponent.
For the first term, 4x^2, raise the coefficient and the variable to the power of 3/14:
4^(3/14) * (x^2)^(3/14) = (2^2)^(3/14) * x^(2 * 3/14) = 2^(2 * 3/14) * x^(6/14) = 2^(6/14) * x^(3/7).
For the second term, y^-1/3, raise the variable to the power of 3/14:
(y^-1/3)^(3/14) = y^(-1/3 * 3/14) = y^(-1/14).
Now, put the simplified terms back together:
(4x^2 y^-1/3)^3/14 = 2^(6/14) * x^(3/7) * y^(-1/14).
So the simplified expression is 2^(6/14) * x^(3/7) * y^(-1/14).
Note: If you need a decimal approximation for the exponent, you can calculate using a calculator.
To simplify the expression (4x^2 y^-1/3)^3/14, we can apply the exponent rules.
First, let's simplify the expression inside the parentheses:
(4x^2 y^-1/3)^3.
To simplify the coefficients, we raise 4 to the power of 3, which is 64.
For x^2, we multiply the exponents: x^2 * 3 = x^6.
For y^-1/3, we multiply the exponent by 3: (-1/3) * 3 = -1.
So, inside the parentheses, we have 64x^6 y^-1.
Now, let's simplify the expression as a whole:
(64x^6 y^-1)^(3/14).
To simplify the exponent, we raise both the numerator and denominator to the power of 3/14:
64^(3/14) * x^(6 * 3/14) * y^(-1 * 3/14).
Using the laws of exponents, we can simplify further:
64^(3/14) = 2^(6 * 3/14) = 2^(18/14) = 2^(9/7).
x^(6 * 3/14) = x^(18/14) = x^(9/7).
y^(-1 * 3/14) = y^(-3/14).
Putting it all together, the simplified expression is:
2^(9/7) * x^(9/7) * y^(-3/14).
Note that all the exponents are positive.