(a^(-2)b)^((1)/(2))times(a^(-3))^((1)/(3))

Jasonbradley

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Jun 16, 2024

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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:

1. 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)

2. Substitute the simplified terms back into the original expression:

   • a^(-1) * b^(1/2) * a^(-1)

3. Apply the second law of exponents to combine the 'a' terms:

   • a^(-1) * a^(-1) = a^(-1 - 1) = a^(-2)

4. 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.