\( r^3 = 144 \). - United Radiology

["Solving ( r^3 = 144 ): A Comprehensive Guide", "Understanding how to solve equations involving exponents is essential in mathematics, physics, engineering, and many applied sciences. One such common problem is solving the equation:", "[
\nr^3 = 144
\n]", "This article provides a clear, step-by-step explanation of how to find the value(s) of ( r ) that satisfy this equation, including both real and complex solutions, and explores practical applications.", "---", "### What Does ( r^3 = 144 ) Mean?", "The equation ( r^3 = 144 ) asks: What number multiplied by itself three times equals 144? In mathematical terms, we are seeking the cube roots of 144.", "---", "### Step-by-Step Solution", "To isolate ( r ), take the cube root of both sides:", "[
\nr = \sqrt[3]{144}
\n]", "Since 144 is not a perfect cube, the cube root cannot be simplified into a whole number. However, we can approximate it or express it in exact or simplified radical form.", "---", "### Simplifying ( \sqrt[3]{144} )", "Begin by factoring 144 into its prime components:", "[
\n144 = 12 \ imes 12 = (2^2 \cdot 3)^2 = 2^4 \cdot 3^2
\n]", "Now, group the exponents to extract complete cube factors:", "[
\n144 = 2^3 \cdot 2 \cdot 3^2
\n]", "Since ( 2^3 = 8 ), we write:", "[
\n\sqrt[3]{144} = \sqrt[3]{8 \cdot (2 \cdot 3^2)} = \sqrt[3]{8} \cdot \sqrt[3]{2 \cdot 9} = 2 \cdot \sqrt[3]{18}
\n]", "Thus, the exact real solution is:", "[
\nr = 2, \sqrt[3]{18}
\n]", "---", "### Finding All Solutions: Real and Complex", "Since every non-zero real number has three distinct complex cube roots, we must consider all solutions.", "The cube roots of 144 can be expressed using complex numbers via De Moivre’s Theorem.", "#### Step 1: Express 144 in polar (complex) form
\nEven though 144 is real, we can represent it in the complex plane as:", "[
\nz = 144 + 0i
\n]", "This corresponds to ( r = 144 ), but more generally, cube roots are located at angles separated by ( \frac{2\pi}{3} ) radians (120°).", "#### Step 2: Use polar form (optional for deeper understanding)", "Write ( 144 = 144 \ ext{ cis}, 0^\circ ), where "cis, \ heta = \cos\ heta + i\sin\ heta ).", "Then the cube roots are:", "[
\nr_k = \sqrt[3]{144} \cdot \ ext{cis}\left( \frac{0^\circ + 360^\circ k}{3} \right) = 2\sqrt[3]{18} \cdot \ ext{cis}\left( 120^\circ k \right), \quad k = 0, 1, 2
\n]", "So the three complex cube roots are:", "- ( r_0 = 2\sqrt[3]{18} ) (real)
\n- ( r_1 = 2\sqrt[3]{18} \left( \cos 120^\circ + i\sin 120^\circ \right) )
\n- ( r_2 = 2\sqrt[3]{18} \left( \cos 240^\circ + i\sin 240^\circ \right) )", "However, because 144 is positive real, only the principal real cube root is typically emphasized unless context demands complex interpretations.", "---", "### Approximate Decimal Value", "To estimate the real solution numerically:", "[
\n\sqrt[3]{144} \approx 5.24 \quad \ ext{(since } 5^3 = 125, ; 5.2^3 = 140.6, ; 5.24^3 \approx 144\ ext{)}
\n]", "More accurately:", "[
\n\sqrt[3]{144} \approx 5.24023
\n]", "Thus, the real solution is approximately:", "[
\nr \approx 5.240
\n]", "---", "### Practical Applications", "Solving equations like ( r^3 = 144 ) arises in numerous fields:", "- Physics: Calculating wave frequencies, radioactive decay half-lives (when exponentiated), or kinetic energy formulations.
\n- Engineering: Determining dimensions related to cubic volumes, such as cubic vaults or reactor cores, where total volume depends on linear dimensions raised to the third power.
\n- Finance: Modeling investments with cubic growth models (though rare), or computing interest compounds in nonlinear systems.
\n- Geometry and Design: Solving for linear edges when volume is given in art, architecture, and 3D modeling.", "---", "### Summary", "The equation ( r^3 = 144 ) has:", "- One real solution:
\n [
\n r = \sqrt[3]{144} = 2\sqrt[3]{18} \approx 5.240
\n ]
\n- Three complex solutions derived via De Moivre’s Theorem, equally spaced on a circle in the complex plane.", "Whether you're solving geometry problems, simulating physical systems, or analyzing exponential growth, understanding how to manipulate and solve cubic root equations is invaluable.", "---", "### Keywords", "- ( r^3 = 144 )
\n- cube root of 144
\n- solve ( r^3 = 144 )
\n- exponentiation and roots
\n- real and complex cube roots
\n- cube root simplification
\n- applications of ( r^3 = k )
\n- algebraic equations, math tutorial", "---", "Start mastering exponential equations today — understanding ( r^3 = 144 ) is a building block!"]