= \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056}

["# Solving \sqrt{21 × 8 × 7 × 6} = √7056: A Step-by-Step Explanation", "When faced with a radical expression like \sqrt{21 \ imes 8 \ imes 7 \ imes 6}, understanding how to simplify it step-by-step can unlock deeper insight into algebra and number properties. In this article, we’ll explore the calculation of \sqrt{21 × 8 × 7 × 6} and verify that it equals \sqrt{7056}, making it easier to appreciate this elegant mathematical simplification.", "## Why Simplify Radicals?", "Before diving into the calculation, it’s worth noting why simplifying radicals matters. Expressing square roots in simplest form enhances clarity, facilitates solving equations, and makes comparison between radical expressions more straightforward. Simplifying \sqrt{7056} reveals that it equals 84, offering a clean, exact value instead of a decimal approximation.", "## Step 1: Calculate the Product Inside the Square Root", "We start with the expression inside the radical:\n[ 21 \ imes 8 \ imes 7 \ imes 6 ]", "Group the factors for easier computation:\nGroup 21 and 7:\n[ 21 \ imes 7 = 147 ]\nGroup 8 and 6:\n[ 8 \ imes 6 = 48 ]", "Now multiply these results:\n[ 147 \ imes 48 ]", "### Multiplying 147 × 48", "We use standard multiplication:\nFirst compute (147 \ imes 40 = 5880)\nThen compute (147 \ imes 8 = 1,176)\nAdd them together:\n[ 5880 + 1,176 = 7,056 ]", "So,\n[ 21 \ imes 8 \ imes 7 \ imes 6 = 7,056 ]", "Thus:\n[ \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} ]", "## Step 2: Simplify √7056", "Now we determine \sqrt{7056}. To simplify, we factor 7056 into its prime components to identify perfect squares.", "### Prime Factorization of 7056", "Start dividing 7056 by small primes:\n7056 is divisible by 2 (even number):\n7056 ÷ 2 = 3528\n3528 ÷ 2 = 1764\n1764 ÷ 2 = 882\n882 ÷ 2 = 441 → Now 441 is not divisible by 2.", "441 is a perfect square:\n441 = 21 × 21 = (3 × 7) × (3 × 7) = 3² × 7²", "So, compiling all:\n7056 = 2⁴ × 3² × 7²", "Now apply the square root:\n[ \sqrt{7056} = \sqrt{2^4 \ imes 3^2 \ imes 7^2} ]\n[ = \sqrt{2^4} \ imes \sqrt{3^2} \ imes \sqrt{7^2} ]\n[ = 2^2 \ imes 3 \ imes 7 ]\n[ = 4 \ imes 3 \ imes 7 = 84 ]", "Thus,\n[ \sqrt{7056} = 84 ]", "## Summary", "We showed that\n[ \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84 ]", "This demonstrates how breaking down a complex radical into simpler prime factors leads to an exact, simplified result—essential in algebra, geometry, and higher mathematics. Whether solving equations or working with geometric areas, understanding radicals helps unlock mathematical fluency and precision.", "Simplify bold:\n\boxed{\sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84}", "### Final Thoughts", "Mastering such computations empowers you to simplify expressions quickly, recognize patterns, and approach complex problems with confidence. Next time you see \sqrt{21 × 8 × 7 × 6}, you’ll know it stretches cleanly to 84—proof that math thrives on clarity and logical steps."]