Simplify the expression \((3x^2y^3)^2 \times (2xy^4)^3\).

["# Simplify the Expression ((3x^2y^3)^2 \ imes (2xy^4)^3 – A Step-by-Step Guide", "When working with algebraic expressions involving exponents, simplification can feel overwhelming — but breaking it down step-by-step makes it much easier. Today, we’ll simplify the expression:", "[\n(3x^2y^3)^2 \ imes (2xy^4)^3\n]", "Whether you're preparing for exams, solving math problems, or just improving your algebraic skills, mastering this expression helps strengthen your foundation. Let’s go through the simplification process clearly and concisely.", "---", "### Step 1: Apply the Power of a Product Rule", "The first step is applying the power of a product rule:\n[\n(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\n]\nHowever, here each base is a product raised to a power, so we’ll apply the power of a power and partial product rules systematically.", "Start by expanding each part separately:", "- ((3x^2y^3)^2)\n- ((2xy^4)^3)", "---", "### Step 2: Simplify Each Component", "#### Simplify ((3x^2y^3)^2)", "- Apply the power to every factor:\n [\n 3^2 \ imes (x^2)^2 \ imes (y^3)^2 = 9 \cdot x^{2 \ imes 2} \cdot y^{3 \ imes 2}\n ]\n- Calculating exponents:\n [\n 9x^4y^6\n ]", "#### Simplify ((2xy^4)^3)", "- Raise each factor to the 3rd power:\n [\n 2^3 \ imes x^3 \ imes (y^4)^3 = 8 \cdot x^3 \cdot y^{4 \ imes 3}\n ]\n- Calculating:\n [\n 8x^3y^{12}\n ]", "---", "### Step 3: Multiply the Simplified Expressions", "Now our expression becomes:", "[\n(9x^4y^6) \ imes (8x^3y^{12})\n]", "Since we’re multiplying like bases, apply the product of powers rule:\n[\na^m \cdot a^n = a^{m+n}\n]", "Multiply the coefficients:\n[\n9 \ imes 8 = 72\n]", "Add the exponents for (x):\n[\nx^4 \cdot x^3 = x^{4+3} = x^7\n]", "Add the exponents for (y):\n[\ny^6 \cdot y^{12} = y^{6+12} = y^{18}\n]", "---", "### Final Simplified Expression", "Combine all the parts:", "[\n72x^7y^{18}\n]", "---", "### Why Simplify Algebraic Expressions?", "Simplification helps reduce complexity, avoid errors in further calculations, improve readability, and prepare solutions for higher-level math such as calculus, derivatives, and integrals.", "---", "### Summary", "To simplify ((3x^2y^3)^2 \ imes (2xy^4)^3):", "1. Apply power to each factor — exponents distribute.\n2. Multiply coefficients.\n3. Combine like bases using exponent rules.", "Final simplified result:\n[\n\boxed{72x^7y^{18}}\n]", "---", "### Key Takeaways", "- Power of a power: ((a^m)^n = a^{m \cdot n})\n- Product of powers: (a^m \cdot a^n = a^{m+n})\n- Distribute exponents when raising products to powers.\n- Multiply coefficients normally and combine exponents carefully.", "Mastering these rules will make complex expressions easier and faster to solve — so practice often and simplify confidently!", "---", "Want to ace algebraic simplification? Check out our next article: Top 10 Tips to Simplify Radicals and Rational Expressions Fast."]