\( 2r^3 = 288 \). - United Radiology

["### Solving ( 2r^3 = 288 ): Step-by-Step Guide", "If you’re working with the equation ( 2r^3 = 288 ), solving for ( r ) is a straightforward process best understood through algebraic manipulation and evaluation. This guide will walk you through step-by-step how to solve this cubic equation and interpret the solution, while also highlighting key terms and methods useful for beginners and students alike.", "---", "### What is the Equation ( 2r^3 = 288 )?", "The equation ( 2r^3 = 288 ) is a simple algebraic expression involving a cubic term. Here, ( r ) is an unknown variable. The goal is to isolate ( r ) and find its value by simplifying the equation.", "---", "### Step 1: Isolate the Cubic Term", "Start by eliminating the coefficient ( 2 ) from the left-hand side. Divide both sides of the equation by 2:", "[
\n2r^3 = 288
\n]", "[
\n\Rightarrow r^3 = \frac{288}{2} = 144
\n]", "Now, you have:", "[
\nr^3 = 144
\n]", "---", "### Step 2: Solve for ( r ) — Taking the Cube Root", "To find ( r ), apply the cube root to both sides of the equation:", "[
\nr = \sqrt[3]{144}
\n]", "This expression represents the real cube root of 144. Since 144 is not a perfect cube, ( r ) is an irrational number, but you can approximate it numerically.", "---", "### Step 3: Approximate the Value", "You can estimate ( \sqrt[3]{144} ) using known cube values:", "- ( 5^3 = 125 )
\n- ( 6^3 = 216 )", "Since ( 125 < 144 < 216 ), the cube root lies between 5 and 6. A better estimate using a calculator:", "[
\n\sqrt[3]{144} \approx 5.24
\n]", "Thus, the solution is approximately:", "[
\nr \approx 5.24
\n]", "---", "### Alternative: Expressing Exact and Approximate Solutions", "- Exact Solution: ( r = \sqrt[3]{144} )
\n- Approximate Solution: ( r \approx 5.24 ) (rounded to two decimal places)", "---", "### Why Solve Equations Like This?", "Understanding how to solve ( 2r^3 = 288 ) builds foundational algebra skills useful in physics, engineering, and computer science, where cubic relationships frequently appear. Whether you are modeling volume, scaling, or financial growth, mastering cube roots and exponents enhances your problem-solving toolkit.", "---", "### Key takeaways:", "- Divide both sides by 2 to eliminate the coefficient: ( r^3 = 144 ).
\n- Take the cube root to isolate ( r ): ( r = \sqrt[3]{144} ).
\n- Approximate numerically when needed: ( r \approx 5.24 ).", "Mastering such equations strengthens your mathematical foundation — unlocking more complex problem-solving ahead!", "---", "### Related Keywords for SEO:", "- Solve ( 2r^3 = 288 )
\n- Cube root of 144
\n- Solve cubic equations step by step
\n- Algebraic solutions for radicals
\n- How to solve ( r^3 = 144 )
\n- Real root and rational approximation of ( r )
\n- Solving equations with powers
\n- Mathematics tutorial: cube roots and exponents", "---", "### Final Note", "Remember, practicing such equations strengthens algebraic fluency. Use online calculators or graphing tools when needed, but always understand each step — true mastery comes from clarity, not just speed.", "---", "Keywords: ( 2r^3 = 288 ), solve ( 2r^3 = 288 ), cube root of 144, algebraic solutions, how to solve cubic equations, mathematics tutorial, exponent rules, real cube root, step-by-step solving.", "---", "If you found this guide helpful, share it with fellow students and reinforce your learning!"]