(4x^2y^5)^2/(2xy^2)^3(3x^3)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Algebraic Expressions: A Step-by-Step Guide

This article will guide you through simplifying the algebraic expression: (4x^2y^5)^2/(2xy^2)^3(3x^3)^2. We'll break down the steps for clarity.

Understanding the Rules

Before we begin, let's recall some essential rules for simplifying expressions with exponents:

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

Simplifying the Expression

Now, let's simplify the expression step-by-step:

1. Apply the Power of a Product rule:

   • (4x^2y^5)^2 = 4^2 * (x^2)^2 * (y^5)^2 = 16x^4y^10
   • (2xy^2)^3 = 2^3 * (x)^3 * (y^2)^3 = 8x^3y^6
   • (3x^3)^2 = 3^2 * (x^3)^2 = 9x^6

2. Substitute the simplified terms back into the original expression:

   • (16x^4y^10) / (8x^3y^6)(9x^6)

3. Apply the Product of Powers rule for the denominator:

   • (16x^4y^10) / (72x^9y^6)

4. Apply the Quotient of Powers rule:

   • 16/72 * x^(4-9) * y^(10-6)

5. Simplify the coefficients and exponents:

   • (2/9) * x^(-5) * y^4

6. Express the negative exponent in the denominator:

   • (2y^4) / (9x^5)

Final Result

The simplified form of the expression (4x^2y^5)^2/(2xy^2)^3(3x^3)^2 is (2y^4) / (9x^5).