Solving for \( w \): \( w^2 = \frac{192}{3} = 64 \).
["Solving for ( w ): A Step-by-Step Guide to Finding the Square Root", "Math problems like solving ( w^2 = \frac{192}{3} = 64 ) are essential building blocks for developing algebraic reasoning and problem-solving skills. This article walks you through solving the equation step by step, clarifying how to simplify expressions, compute square roots, and arrive at the final value of ( w ).", "---", "### Understanding the Equation", "We begin with the equation:\n[ w^2 = \frac{192}{3} = 64 ]", "At first glance, the equation sets a squared unknown ( w ) equal to 64. Solving for ( w ) involves taking the square root of both sides, a fundamental algebraic operation critical for simplifying radical expressions and equations.", "---", "### Step 1: Simplify the Right-Hand Side", "The fraction ( \frac{192}{3} ) is already simplified, since 3 divides 192 evenly:\n[ \frac{192}{3} = 64 ]\nSo the equation becomes:\n[ w^2 = 64 ]", "---", "### Step 2: Take the Square Root of Both Sides", "To solve for ( w ), take the square root of both sides:\n[ w = \pm \sqrt{64} ]", "Since ( \sqrt{64} = 8 ), we have:\n[ w = \pm 8 ]", "---", "### What Does This Mean?", "The solution ( w = 8 ) and ( w = -8 ) means both values satisfy the original equation:\n- ( (8)^2 = 64 )\n- ( (-8)^2 = 64 )", "Squaring a negative number yields a positive result, which explains why both roots are valid.", "---", "### Why This Matters", "This simple example illustrates key algebraic principles:\n- Equations involving squares require careful consideration of both positive and negative roots.\n- Simplifying fractions before solving makes computations more efficient.\n- Square roots like ( \sqrt{64} ) consistently yield 8 — understanding this anchor helps build confidence in more complex problems.", "---", "### Final Answer", "Solving for ( w ) in ( w^2 = \frac{192}{3} = 64 ) gives:\n[\n\boxed{w = \pm 8}\n]", "---", "Bonus Tips:", "- Always simplify fractions before substituting into equations.\n- Remember, ( w^2 = a ) implies ( w = \sqrt{a} ) or ( w = -\sqrt{a} ).\n- Practice with different values to master sign considerations.", "---", "Understanding how to solve equations like ( w^2 = 64 ) sets a strong foundation for algebra, calculus, and beyond — making math less daunting and more empowering!"]
