(2x^-2y^3/3xy^-4)^4

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (2x^-2y^3/3xy^-4)^4

This article will guide you through the process of simplifying the expression (2x^-2y^3/3xy^-4)^4.

Understanding the Rules

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

• Product of powers: x^m * x^n = x^(m+n)
• Quotient of powers: x^m / x^n = x^(m-n)
• Power of a product: (xy)^n = x^n * y^n
• Power of a quotient: (x/y)^n = x^n / y^n
• Negative exponent: x^-n = 1/x^n

Step-by-Step Simplification

1. Apply the power of a quotient rule:

   (2x^-2y^3/3xy^-4)^4 = (2^4 * x^-8 * y^12) / (3^4 * x^4 * y^-16)

2. Simplify the coefficients:

   = (16x^-8y^12) / (81x^4y^-16)

3. Apply the quotient of powers rule for x and y:

   = (16/81) * x^(-8-4) * y^(12+16)

4. Simplify the exponents:

   = (16/81) * x^-12 * y^28

5. Apply the negative exponent rule to x:

   = (16/81) * (1/x^12) * y^28

6. Combine the terms:

   = 16y^28 / (81x^12)

Conclusion

Therefore, the simplified expression of (2x^-2y^3/3xy^-4)^4 is 16y^28 / (81x^12). By applying the rules of exponents and simplifying each step, we can arrive at a more concise and understandable form.