Simplifying the Expression: (a^(2)b)^((1)/(2)) times (a^(3))^((1)/(3))
This article will explore the process of simplifying the expression: (a^(2)b)^((1)/(2)) times (a^(3))^((1)/(3)). We will utilize the laws of exponents to break down the expression and arrive at a simplified form.
Understanding the Laws of Exponents
To simplify the expression, we need to recall the following laws of exponents:
 (a^m)^n = a^(m*n): This states that when raising a power to another power, we multiply the exponents.
 a^m * a^n = a^(m+n): This states that when multiplying powers with the same base, we add the exponents.
 a^(m) = 1/a^m: This states that a negative exponent indicates the reciprocal of the base raised to the positive exponent.
Simplifying the Expression
Let's break down the simplification step by step:

Apply the first law of exponents to both terms:
 (a^(2)b)^((1)/(2)) = a^(2 * (1/2)) * b^(1/2) = a^(1) * b^(1/2)
 (a^(3))^((1)/(3)) = a^(3 * (1/3)) = a^(1)

Substitute the simplified terms back into the original expression:
 a^(1) * b^(1/2) * a^(1)

Apply the second law of exponents to combine the 'a' terms:
 a^(1) * a^(1) = a^(1  1) = a^(2)

Apply the third law of exponents to express the final result with positive exponents:
 a^(2) * b^(1/2) = 1/(a^2 * b^(1/2))
Conclusion
Therefore, the simplified form of the expression (a^(2)b)^((1)/(2)) times (a^(3))^((1)/(3)) is 1/(a^2 * b^(1/2)). This process demonstrates the importance of understanding and applying the laws of exponents for effective simplification of mathematical expressions.