(3x^-4)^2(5x^-2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (3x^-4)^2(5x^-2)

This article will guide you through simplifying the expression (3x^-4)^2(5x^-2). We'll break down the steps involved, explaining the key concepts in detail.

Understanding the Properties of Exponents

Before we begin simplifying, let's review some crucial properties of exponents:

• Product of powers: When multiplying powers with the same base, we add the exponents.
  • Example: x^m * x^n = x^(m+n)
• Power of a product: When raising a product to a power, we raise each factor to that power.
  • Example: (xy)^n = x^n * y^n
• Power of a power: When raising a power to another power, we multiply the exponents.
  • Example: (x^m)^n = x^(m*n)

Simplifying the Expression

1. Simplify the square: Apply the "power of a product" rule to (3x^-4)^2:

   • (3x^-4)^2 = 3^2 * (x^-4)^2 = 9x^-8

2. Apply the "power of a power" rule:

   • (x^-4)^2 = x^(-4*2) = x^-8

3. Combine the simplified terms:

   • 9x^-8 * 5x^-2

4. Apply the "product of powers" rule:

   • 9 * 5 * x^(-8-2) = 45x^-10

5. Express the answer with positive exponents: Using the rule x^-n = 1/x^n, we get:

   • 45x^-10 = 45/x^10

Conclusion

Therefore, the simplified expression of (3x^-4)^2(5x^-2) is 45/x^10. By understanding the fundamental properties of exponents, we can easily manipulate and simplify complex expressions.