%5E5(4xy%5E-3)%5E-7.jpeg "(12x^2y^-2)^5(4xy^-3)^-7")
Simplifying Expressions with Exponents
This article will explore the simplification of the expression (12x²y⁻²)^5(4xy⁻³)^-7. We will use the rules of exponents to break down the expression and arrive at a simplified form.
Understanding the Rules of Exponents
Before we begin, let's recall some key rules of exponents that will be crucial in this process:
- Product of powers: xᵃ * xᵇ = xᵃ⁺ᵇ
- Power of a product: (xy)ᵃ = xᵃ * yᵃ
- Power of a power: (xᵃ)ᵇ = xᵃᵇ
- Negative exponent: x⁻ᵃ = 1/xᵃ
Simplifying the Expression
-
Apply the power of a power rule:
- (12x²y⁻²)^5 = 12⁵ * x²⁵ * y⁻¹⁰
- (4xy⁻³)^-7 = 4⁻⁷ * x⁻⁷ * y²¹
-
Simplify the coefficients:
- 12⁵ = 248,832
- 4⁻⁷ = 1/16,384
-
Combine the terms:
- (12x²y⁻²)^5(4xy⁻³)^-7 = (248,832 * x²⁵ * y⁻¹⁰) * (1/16,384 * x⁻⁷ * y²¹)
-
Apply the product of powers rule:
- 248,832 * (1/16,384) * x²⁵ * x⁻⁷ * y⁻¹⁰ * y²¹ = 15.25 * x¹⁸ * y¹¹
Final Result
Therefore, the simplified form of the expression (12x²y⁻²)^5(4xy⁻³)^-7 is 15.25x¹⁸y¹¹.
Conclusion
By carefully applying the rules of exponents, we successfully simplified the complex expression into a much more manageable form. This process highlights the importance of understanding these rules and utilizing them efficiently to solve mathematical problems.