Answer:
To simplify the expression \((27x^{-3}y^6)^{-1/3}(xy^{1/3})\), you can use the properties of exponents and the rules of exponentiation.
Let's break it down step by step:
1. First, simplify the expression inside the parentheses:
\((27x^{-3}y^6)^{-1/3} = 27^{-1/3}x^{-3*(-1/3)}y^{6*(-1/3)}\)
2. Simplify the exponents:
\(27^{-1/3} = (3^3)^{-1/3} = 3^{3*(-1/3)} = 3^{-1} = 1/3\)
\(x^{-3*(-1/3)} = x^1 = x\)
\(y^{6*(-1/3)} = y^{-2}\)
3. Substitute these values back into the expression:
\((27x^{-3}y^6)^{-1/3} = (1/3)xy^{-2}(xy^{1/3})\)
4. Now, multiply the terms with the same base (x and y):
\(1/3 * x * x * y^{-2} * y^{1/3}\)
5. Use the properties of exponents to simplify further:
\(1/3 * x^2 * y^{-2 + 1/3}\)
6. Combine the exponents:
\(1/3 * x^2 * y^{-5/3}\)
So, the simplified expression is \(\frac{1}{3}x^2y^{-5/3}\).
Explanation: