(2x^-3)^2(4x^-4)

Jasonbradley

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Jun 16, 2024

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Simplifying Expressions with Exponents

This article will guide you through the process of simplifying the expression (2x^-3)^2(4x^-4). We will use the rules of exponents to make the expression easier to understand and work with.

Understanding the Rules of Exponents

Before we begin simplifying, let's refresh our memory on the key rules of exponents:

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

Simplifying the Expression

Now, let's apply these rules to our expression:

1. Simplify (2x^-3)^2:

   • Using the power of a product rule: (2x^-3)^2 = 2^2 * (x^-3)^2
   • Using the power of a power rule: 2^2 * (x^-3)^2 = 4 * x^(-3*2) = 4x^-6

2. Simplify 4x^-4:

   • This part already looks simplified, but we can rewrite it using the negative exponent rule: 4x^-4 = 4/x^4

3. Combine the simplified parts:

   • (2x^-3)^2(4x^-4) = 4x^-6 * 4/x^4

4. Use the product of powers rule to simplify further:

   • 4x^-6 * 4/x^4 = 16 * x^(-6+4) = 16x^-2

5. Rewrite using the negative exponent rule:

   • 16x^-2 = 16/x^2

Conclusion

The simplified form of the expression (2x^-3)^2(4x^-4) is 16/x^2. By applying the rules of exponents, we can break down complex expressions into simpler forms, making them easier to understand and manipulate.