Compute the square of \( \sqrt{18} + \sqrt{8} \) and simplify the result.
["Compute the Square of ( \sqrt{18} + \sqrt{8} ) and Simplify the Result", "Understanding how to compute the square of an expression involving square roots is essential in algebra and simplifying radical expressions. In this article, we’ll compute ( \left( \sqrt{18} + \sqrt{8} \right)^2 ), simplify the result, and explore how radicals simplify—especially when dealing with square roots.", "---", "### Step 1: Expand the Square Expression", "We begin by applying the algebraic identity for the square of a binomial:", "[\n(a + b)^2 = a^2 + 2ab + b^2\n]", "Here, ( a = \sqrt{18} ) and ( b = \sqrt{8} ). Substituting these into the formula:", "[\n\left( \sqrt{18} + \sqrt{8} \right)^2 = (\sqrt{18})^2 + 2\sqrt{18}\sqrt{8} + (\sqrt{8})^2\n]", "---", "### Step 2: Simplify Each Term", "Now simplify each part:", "- ( (\sqrt{18})^2 = 18 )\n- ( (\sqrt{8})^2 = 8 )\n- ( 2\sqrt{18} \cdot \sqrt{8} = 2\sqrt{18 \ imes 8} = 2\sqrt{144} )", "So the expression becomes:", "[\n18 + 8 + 2\sqrt{144}\n]", "---", "### Step 3: Simplify the Square Roots", "Now simplify ( \sqrt{144} ):", "[\n\sqrt{144} = 12 \quad \ ext{since } 12^2 = 144\n]", "Therefore:", "[\n2\sqrt{144} = 2 \ imes 12 = 24\n]", "---", "### Step 4: Add All Components", "Now sum all simplified terms:", "[\n18 + 8 + 24 = 50\n]", "---", "### Final Result", "Thus, the square of ( \sqrt{18} + \sqrt{8} ) simplifies to:", "[\n\left( \sqrt{18} + \sqrt{8} \right)^2 = \boxed{50}\n]", "---", "### Key Takeaways for Simplifying Square Roots", "- Always reduce square roots to their simplest radical form:\n ( \sqrt{18} = \sqrt{9 \ imes 2} = 3\sqrt{2} )\n ( \sqrt{8} = \sqrt{4 \ imes 2} = 2\sqrt{2} )\n- Use the identity ( (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} ) for easy computation.\n- Recognize perfect squares inside radicals to simplify more efficiently.", "This method applies broadly to simplify expressions involving sums, differences, or products of square roots—and ensures your answers are clean, precise, and presented in simplest radical form."]
