Multiply both sides by $ 6\sqrt{2} $: - United Radiology

["# Understanding How to Multiply Both Sides by $ 6\sqrt{2} $: A Step-by-Step Guide", "When solving equations in algebra, one common operation is multiplying both sides by a scalar to isolate the variable. Today, we’ll explore the important step of multiplying both sides by $ 6\sqrt{2} $—what it means, how to do it correctly, and why it’s essential for solving equations efficiently.", "## Why Multiply Both Sides by $ 6\sqrt{2} $?", "Multiplying both sides of an equation by a positive scalar (like $ 6\sqrt{2} $) preserves equality while transforming the equation into a simpler form. This process is especially useful when dealing with equations that involve irrational coefficients or need rationalization, making it easier to simplify and solve.", "Consider an example equation where $ x $ is multiplied by $ 6\sqrt{2} $:", "[
\nx = \frac{5}{6\sqrt{2}}
\n]", "To isolate $ x $, we multiply both sides by $ 6\sqrt{2} $:", "[
\n6\sqrt{2} \cdot x = 6\sqrt{2} \cdot \frac{5}{6\sqrt{2}}
\n]", "The right-hand side simplifies:", "[
\n6\sqrt{2} \cdot \frac{5}{6\sqrt{2}} = 5
\n]", "So the equation becomes:", "[
\n6\sqrt{2} \cdot x = 5
\n]", "Now it’s clear that:", "[
\nx = \frac{5}{6\sqrt{2}}
\n]", "But often, we want to eliminate the radical from the denominator. To do this effectively, we can rationalize the denominator.", "## Step 1: Multiply by $ 6\sqrt{2} $", "Starting with:", "[
\nx = \frac{5}{6\sqrt{2}}
\n]", "Multiply both sides by $ 6\sqrt{2} $:", "[
\n6\sqrt{2} \cdot x = 6\sqrt{2} \cdot \frac{5}{6\sqrt{2}}
\n]", "This simplifies directly to:", "[
\n6\sqrt{2} \cdot x = 5
\n]", "## Step 2: Rationalize the Expression", "Now simplify $ \frac{5}{6\sqrt{2}} $ by multiplying numerator and denominator by $ \sqrt{2} $:", "[
\n\frac{5}{6\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{6 \cdot 2} = \frac{5\sqrt{2}}{12}
\n]", "So the fully simplified solution is:", "[
\nx = \frac{5\sqrt{2}}{12}
\n]", "## Conclusion", "Multiplying both sides by $ 6\sqrt{2} $ is a powerful technique to eliminate coefficients and simplify equations containing radicals. This operation not only isolates the variable but also opens the door to rationalizing denominators for cleaner, more interpretable results. Whether you're solving algebraic expressions or preparing equations for advanced computation, understanding how and why to multiply by $ 6\sqrt{2} $ is a valuable skill in mathematics.", "---", "Key takeaways:", "- Multiplying both sides by $ 6\sqrt{2} $ clears scalar multipliers involving radicals.
\n- This prepares equations for simplification and rationalization.
\n- The rationalized solution is $ x = \frac{5\sqrt{2}}{12} $, easier to work with in further calculations.", "Whether you're a student mastering algebra or a teacher explaining key techniques, multiplying both sides carefully by $ 6\sqrt{2} $ remains a fundamental step toward solving equations accurately and cleanly.", "# Algebra Tips | Learning Math | Equation Simplification #"]