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

3 min read Jun 16, 2024
(3xy^4)^2(2x^2y)^3

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

This article will guide you through simplifying the expression (3xy^4)^2(2x^2y)^3. We'll break down the steps and explain the properties of exponents involved.

Understanding the Properties of Exponents

To simplify this expression, we'll utilize the following properties of exponents:

  • Product of Powers: x^m * x^n = x^(m+n)
  • Power of a Product: (xy)^n = x^n * y^n
  • Power of a Power: (x^m)^n = x^(m*n)

Simplifying the Expression Step-by-Step

  1. Distribute the exponents: Apply the "Power of a Product" property to both terms:

    • (3xy^4)^2 = 3^2 * x^2 * (y^4)^2
    • (2x^2y)^3 = 2^3 * (x^2)^3 * y^3

  2. Simplify the exponents: Apply the "Power of a Power" property:

    • 3^2 * x^2 * (y^4)^2 = 9x^2y^8
    • 2^3 * (x^2)^3 * y^3 = 8x^6y^3

  3. Combine the terms: Now, we have the expression: 9x^2y^8 * 8x^6y^3. Apply the "Product of Powers" property:

    • 9x^2y^8 * 8x^6y^3 = (9 * 8) * (x^2 * x^6) * (y^8 * y^3)

  4. Final simplification:

    • (9 * 8) * (x^2 * x^6) * (y^8 * y^3) = 72x^8y^11

Conclusion

By applying the properties of exponents, we have successfully simplified the expression (3xy^4)^2(2x^2y)^3 to 72x^8y^11. Remember to always follow the order of operations (PEMDAS/BODMAS) and break down complex expressions into simpler steps for easier understanding and calculation.

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