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(a^2 B^-3)^4/(a^2 B^6)^-1

2 min read Jun 16, 2024
(a^2 B^-3)^4/(a^2 B^6)^-1

Simplifying Exponential Expressions: (a^2 b^-3)^4/(a^2 b^6)^-1

This article will guide you through simplifying the complex exponential expression: (a^2 b^-3)^4/(a^2 b^6)^-1.

Understanding the Rules

Before we dive into the simplification process, let's refresh our memory on some key exponent rules:

  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n)
  • Power of a power: (a^m)^n = a^(m*n)
  • Negative exponent: a^-n = 1/a^n

Applying the Rules

Let's break down the simplification step-by-step:

  1. Apply the power of a power rule:

    • (a^2 b^-3)^4 = a^(24) b^(-34) = a^8 b^-12
    • (a^2 b^6)^-1 = a^(2*-1) b^(6*-1) = a^-2 b^-6

  2. Substitute the simplified terms into the original expression:

    • (a^8 b^-12) / (a^-2 b^-6)

  3. Apply the quotient of powers rule:

    • a^(8 - (-2)) b^(-12 - (-6))

  4. Simplify:

    • a^10 b^-6

  5. Apply the negative exponent rule to b^-6:

    • a^10 / b^6

Final Answer

Therefore, the simplified form of (a^2 b^-3)^4/(a^2 b^6)^-1 is a^10 / b^6.

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