A = \sqrt{28 \times 21 \times 4 \times 3}

["Solving A = √(28 × 21 × 4 × 3): Step-by-Step Breakdown", "If you’ve stumbled across the expression A = √(28 × 21 × 4 × 3), you’re likely trying to simplify square roots or solve a mathematical problem involving radicals. Whether you’re a student encountering this in algebra or simply curious about radical simplification, this article walks you through solving A = √(28 × 21 × 4 × 3) step-by-step.", "---", "### What is the Expression A = √(28 × 21 × 4 × 3)?", "This equation defines A as the square root of the product of four numbers: 28, 21, 4, and 3. At first glance, multiplying these all together may seem tedious, but mathematical shortcuts and prime factorization simplify the process significantly. Understanding how to simplify square roots is essential in algebra, geometry, and advanced math.", "---", "### Step 1: Multiply the Numbers Inside the Square Root", "Begin by calculating the product under the radical:", "[\nA = \sqrt{28 \ imes 21 \ imes 4 \ imes 3}\n]", "Multiply step-by-step:", "- First: (28 \ imes 21 = 588)\n- Then: (4 \ imes 3 = 12)\n- Now: (588 \ imes 12)", "To simplify:", "[\n588 \ imes 12 = (588 \ imes 10) + (588 \ imes 2) = 5880 + 1176 = 7056\n]", "So,", "[\nA = \sqrt{7056}\n]", "---", "### Step 2: Prime Factorization for Simplification", "While multiplying gives us 7056, simplifying square roots is easier using prime factorization. Break each number into prime factors:", "- (28 = 2^2 \ imes 7)\n- (21 = 3 \ imes 7)\n- (4 = 2^2)\n- (3 = 3)", "Now multiply:", "[\n28 \ imes 21 \ imes 4 \ imes 3 = (2^2 \ imes 7) \ imes (3 \ imes 7) \ imes (2^2) \ imes 3\n]", "Combine like terms:", "[\n= 2^2 \ imes 2^2 \ imes 3 \ imes 3 \ imes 7 \ imes 7 = 2^4 \ imes 3^2 \ imes 7^2\n]", "---", "### Step 3: Apply Square Root Using Exponent Rules", "Recall that:", "[\n\sqrt{a^m} = a^{m/2}\n]", "So,", "[\n\sqrt{2^4 \ imes 3^2 \ imes 7^2} = 2^{4/2} \ imes 3^{2/2} \ imes 7^{2/2} = 2^2 \ imes 3^1 \ imes 7^1\n]", "Calculate:", "[\n= 4 \ imes 3 \ imes 7 = 84\n]", "---", "### Final Result", "[\n\boxed{A = \sqrt{28 \ imes 21 \ imes 4 \ imes 3} = \sqrt{7056} = 84}\n]", "---", "### Why This Simplification Matters", "This type of radical simplification appears in geometry (e.g., diagonal lengths), physics (wave amplitude), and engineering when solving equations involving square roots. Breaking down products into prime factors allows efficient computation and clearer expressions, essential for advanced problem-solving.", "---", "### Quick Summary", "- Multiply step-by-step or factor into primes\n- Prime factorization simplifies rearranging square roots\n- (\sqrt{a \ imes b} = \sqrt{a} \ imes \sqrt{b}) (if helpful)\n- Prime powers allow reduction: (\sqrt{p^m} = p^{m/2})\n- Final simplified radical: A = 84", "---", "### Need More Help?", "Mastering radicals unlocks more complex algebra and calculus. Practice simplifying different radicals—you’ll soon find products like this become quick and intuitive!", "---", "Keywords:\nA = √(28 × 21 × 4 × 3), radical simplification, square root calculation, prime factorization, algebra speed, math tips, solve square root, simplify square root, math problem solution", "Meta Description:\nLearn how to solve A = √(28 × 21 × 4 × 3) step-by-step using prime factorization and square root rules. Simplify radicals for algebra success."]