= \sqrt{21 \cdot 8 \cdot 42} = \sqrt{7056} = 84 \text{ cm}^2

["How to Solve √(21 × 8 × 42) = √7056 = 84 cm²: A Step-by-Step Breakdown", "Understanding how to simplify expressions like (\sqrt{21 \cdot 8 \cdot 42} = \sqrt{7056} = 84), especially in the context of area calculations in cm², helps build strong foundations in algebra and geometry. In this article, we’ll walk through the process of evaluating (\sqrt{21 \cdot 8 \cdot 42}), simplifying it correctly, and recognizing that the area equals 84 cm².", "---", "### Step 1: Understand the Expression", "We start with:", "[\n\sqrt{21 \cdot 8 \cdot 42}\n]", "Rather than grabbing a calculator immediately, it’s better to simplify the expression inside the square root first.", "---", "### Step 2: Simplify the Product Inside the Square Root", "Rather than multiplying all three numbers directly, we look for factors and simplifications:", "Compute:\n[\n21 \ imes 8 \ imes 42\n]", "But instead of fully expanding:", "[\n21 = 3 \ imes 7,\quad 8 = 2^3,\quad 42 = 2 \ imes 3 \ imes 7\n]", "So,", "[\n21 \cdot 8 \cdot 42 = (3 \cdot 7) \cdot (2^3) \cdot (2 \cdot 3 \cdot 7)\n]", "Now combine like terms:", "- Powers of 2: (2^3 \cdot 2 = 2^4)\n- Powers of 3: (3 \cdot 3 = 3^2)\n- Powers of 7: (7 \cdot 7 = 7^2)", "Thus,", "[\n21 \cdot 8 \cdot 42 = 2^4 \ imes 3^2 \ imes 7^2\n]", "Now take the square root:", "[\n\sqrt{21 \cdot 8 \cdot 42} = \sqrt{2^4 \cdot 3^2 \cdot 7^2}\n]", "---", "### Step 3: Apply Square Root Properties", "Using the property (\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}), we split:", "[\n\sqrt{2^4} \cdot \sqrt{3^2} \cdot \sqrt{7^2}\n]", "Recall (\sqrt{n^2} = n) for positive (n), so:", "[\n\sqrt{2^4} = 2^2 = 4,\quad \sqrt{3^2} = 3,\quad \sqrt{7^2} = 7\n]", "Multiply:", "[\n4 \cdot 3 \cdot 7 = 12 \cdot 7 = 84\n]", "So,", "[\n\sqrt{21 \cdot 8 \cdot 42} = 84\n]", "---", "### Step 4: Interpret the Result in Area Units", "Since this square root simplifies to 84, and the context mentions cm², it follows that:", "[\n\sqrt{21 \cdot 8 \cdot 42} , \ ext{cm}^2 = 84 , \ ext{cm}^2\n]", "This means the area of a shape (like a rectangle or geometric figure) with dimensions derived from the product 21 × 8 × 42 cm dimensions gives a total area of 84 cm².", "---", "### Why This Simplification Matters", "Simplifying nested square roots like (\sqrt{a \cdot b \cdot c}) without expanding manually is a key algebra skill. It avoids computational errors and helps recognize exact values beneath approximate decimal results.", "In practical terms, expressing area as 84 cm² makes it easier to work with in design, construction, or physics, where precise cubic or square measurements are essential.", "---", "### Final Summary", "- (\sqrt{21 \cdot 8 \cdot 42} = \sqrt{7056})\n- Prime factorization reveals (2^4 \cdot 3^2 \cdot 7^2)\n- Square root simplifies to (2^2 \cdot 3 \cdot 7 = 84)\n- Therefore, (\sqrt{21 \cdot 8 \cdot 42} = 84 , \ ext{cm}^2)", "Mastering these steps empowers you to tackle complex expressions with confidence and accuracy. Whether you're solving math problems or calculating real-world areas, understanding how to simplify under a radical is invaluable.", "---", "Keywords: square root simplification, √(21 × 8 × 42), √7056 = 84 cm², algebra, geometry, area calculation, exact values, mathematical simplification", "Meta Description: Learn how to simplify √(21 × 8 × 42) = √7056 = 84 cm² using prime factorization and radical properties. Step-by-step breakdown with area context."]