(4x^3y^6)^-2+(2x^2y^4)^-3

Jasonbradley

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Jun 16, 2024

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Simplifying Exponential Expressions

This article will guide you through the process of simplifying the expression (4x³y⁶)^-2 + (2x²y⁴)^-3.

Understanding the Properties of Exponents

To simplify this expression, we need to recall the following rules of exponents:

• Product of Powers: (a^m) * (a^n) = a^(m+n)
• Power of a Product: (a*b)^m = a^m * b^m
• Power of a Power: (a^m)^n = a^(m*n)
• Negative Exponent: a^-m = 1/a^m

Step-by-Step Simplification

1. Apply the Power of a Power Rule:

   • (4x³y⁶)^-2 = 4^-2 * (x³)^-2 * (y⁶)^-2 = 1/16 * x^-6 * y^-12
   • (2x²y⁴)^-3 = 2^-3 * (x²)^-3 * (y⁴)^-3 = 1/8 * x^-6 * y^-12

2. Combine the Simplified Terms:

   • (1/16 * x^-6 * y^-12) + (1/8 * x^-6 * y^-12)

3. Use the Negative Exponent Rule:

   • (1/16 * 1/x⁶ * 1/y¹²) + (1/8 * 1/x⁶ * 1/y¹²)

4. Simplify the Expression:

   • (1/16x⁶y¹²) + (1/8x⁶y¹²) = (1 + 2) / (16x⁶y¹²) = 3 / (16x⁶y¹²)

Conclusion

Therefore, the simplified form of the expression (4x³y⁶)^-2 + (2x²y⁴)^-3 is 3 / (16x⁶y¹²). Remember to always follow the order of operations and apply the appropriate exponent rules when simplifying expressions.