(2a^3b^4)^3/(2ab^2)^5

Jasonbradley

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Jun 16, 2024

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Simplifying Algebraic Expressions: (2a^3b^4)^3 / (2ab^2)^5

This article will guide you through simplifying the algebraic expression (2a^3b^4)^3 / (2ab^2)^5. We'll break down the steps using the rules of exponents and demonstrate how to arrive at the simplified form.

Understanding the Rules of Exponents

Before we begin, let's review some key exponent rules:

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

Simplifying the Expression

1. Apply the Power of a Product Rule:

   • (2a^3b^4)^3 = 2^3 * (a^3)^3 * (b^4)^3 = 8a^9b^12
   • (2ab^2)^5 = 2^5 * a^5 * (b^2)^5 = 32a^5b^10

2. Substitute the simplified terms into the original expression:

   • (2a^3b^4)^3 / (2ab^2)^5 = 8a^9b^12 / 32a^5b^10

3. Apply the Quotient of Powers Rule:

   • 8a^9b^12 / 32a^5b^10 = (8/32) * (a^(9-5)) * (b^(12-10)) = (1/4) * a^4 * b^2

4. Final Simplified Expression:

   • (2a^3b^4)^3 / (2ab^2)^5 = a^4b^2 / 4

Conclusion

By applying the rules of exponents, we successfully simplified the expression (2a^3b^4)^3 / (2ab^2)^5 to a^4b^2 / 4. Remember to always work step-by-step, paying attention to the specific rules governing each operation. This approach ensures accuracy and clarity when handling complex algebraic expressions.