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(a^2b^3)^-2(a^5b^4)^-3

2 min read Jun 16, 2024
(a^2b^3)^-2(a^5b^4)^-3

Simplifying Expressions with Negative Exponents

This article will guide you through simplifying the expression (a^2b^3)^-2(a^5b^4)^-3.

Understanding the Rules

Before we begin, let's refresh our memory on some key exponent rules:

  • Power of a product: (ab)^n = a^n * b^n
  • Power of a power: (a^m)^n = a^(m*n)
  • Negative exponent: a^-n = 1/a^n

Applying the Rules

  1. Apply the power of a power rule:

    • (a^2b^3)^-2 = a^(2*-2) * b^(3*-2) = a^-4 * b^-6
    • (a^5b^4)^-3 = a^(5*-3) * b^(4*-3) = a^-15 * b^-12

  2. Combine the results:

    • a^-4 * b^-6 * a^-15 * b^-12

  3. Apply the product of powers rule:

    • a^(-4-15) * b^(-6-12) = a^-19 * b^-18

  4. Apply the negative exponent rule:

    • 1/a^19 * 1/b^18

  5. Simplify further:

    • 1 / (a^19 * b^18)

Conclusion

Therefore, the simplified form of (a^2b^3)^-2(a^5b^4)^-3 is 1 / (a^19 * b^18).

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