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(-6x^4y^6)^2/(-4x^-3y^5)^3

2 min read Jun 16, 2024
(-6x^4y^6)^2/(-4x^-3y^5)^3

Simplifying the Expression: (-6x^4y^6)^2/(-4x^-3y^5)^3

This problem involves simplifying an expression with exponents and negative exponents. Let's break down the steps:

Understanding the Rules of Exponents

  • Power of a product: (ab)^n = a^n * b^n
  • Power of a quotient: (a/b)^n = a^n / b^n
  • Negative exponents: a^-n = 1/a^n
  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n)

Simplifying the Expression

  1. Apply the power of a product rule to both numerator and denominator: (-6x^4y^6)^2 = (-6)^2 * (x^4)^2 * (y^6)^2 = 36x^8y^12 (-4x^-3y^5)^3 = (-4)^3 * (x^-3)^3 * (y^5)^3 = -64x^-9y^15

  2. Apply the power of a quotient rule: (36x^8y^12) / (-64x^-9y^15)

  3. Simplify by applying the quotient of powers rule: 36/(-64) * x^(8-(-9)) * y^(12-15) = -9/16 * x^17 * y^-3

  4. Apply the negative exponent rule to y^-3: -9/16 * x^17 * 1/y^3

Final Result

The simplified form of the expression (-6x^4y^6)^2/(-4x^-3y^5)^3 is -9x^17 / (16y^3).

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