Calculate: \(x^3 = \left(\frac{47}{11}\right)^{3/2}\), \(6 \times \left(\frac{47}{11}\right)^{3/2}\).
["Title: How to Calculate (x^3 = \left(\frac{47}{11}\right)^{3/2}) and Compute (6 \ imes \left(\frac{47}{11}\right)^{3/2}) – Step-by-Step Guide", "---", "Introduction\nSolving exponential and fractional power equations is a common challenge in algebra and applied mathematics. In this article, we walk through the step-by-step process of calculating (x^3 = \left(\frac{47}{11}\right)^{3/2}), then compute (6 \ imes \left(\frac{47}{11}\right)^{3/2}), making complex math accessible and practical.", "---", "### Understanding the Equation: (x^3 = \left(\frac{47}{11}\right)^{3/2})", "The equation combines exponentiation and roots. To isolate (x), we take the cube root of both sides:", "[\nx = \left(\left(\frac{47}{11}\right)^{3/2}\right)^{1/3}\n]", "Using the power rule ((a^m)^n = a^{m \cdot n}), simplify the exponent:", "[\nx = \left(\frac{47}{11}\right)^{(3/2) \cdot (1/3)} = \left(\frac{47}{11}\right)^{1/2} = \sqrt{\frac{47}{11}}\n]", "### Simplify the Square Root Expression", "[\nx = \sqrt{\frac{47}{11}} = \frac{\sqrt{47}}{\sqrt{11}}\n]", "Rationalizing the denominator:", "[\nx = \frac{\sqrt{47} \cdot \sqrt{11}}{11} = \frac{\sqrt{517}}{11}\n]", "So, the exact value of (x) is (\frac{\sqrt{517}}{11}).", "---", "### Now Compute (6 \ imes \left(\frac{47}{11}\right)^{3/2})", "Let’s simplify (6 \ imes \left(\frac{47}{11}\right)^{3/2}) step by step.", "We write:", "[\n6 \ imes \left(\frac{47}{11}\right)^{3/2} = 6 \ imes \left(\sqrt{\frac{47}{11}}\right)^3 = 6 \ imes \left(\sqrt{\frac{47}{11}}\right) \cdot \left(\sqrt{\frac{47}{11}}\right)^2\n]", "But more directly, since (\left(\frac{47}{11}\right)^{3/2} = \frac{47}{11} \cdot \sqrt{\frac{47}{11}}), multiply by 6:", "[\n6 \ imes \left(\frac{47}{11}\right)^{3/2} = 6 \cdot \frac{47}{11} \cdot \sqrt{\frac{47}{11}} = \frac{282}{11} \cdot \sqrt{\frac{47}{11}}\n]", "Alternatively, express everything as radicals and fractions:", "[\n6 \ imes \left(\frac{47}{11}\right)^{3/2} = 6 \cdot \left(\frac{47^{3/2}}{11^{3/2}}\right) = 6 \cdot \frac{\sqrt{47^3}}{\sqrt{11^3}} = 6 \cdot \frac{47\sqrt{47}}{11\sqrt{11}}\n]", "Rationalizing the denominator:", "[\n= \frac{6 \cdot 47\sqrt{47}}{11 \cdot \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{282\sqrt{47} \cdot \sqrt{11}}{11 \cdot 11} = \frac{282\sqrt{517}}{121}\n]", "---", "### Final Values", "- Exact solution for (x^3 = \left(\frac{47}{11}\right)^{3/2}):\n [\n x = \frac{\sqrt{517}}{11}\n ]", "- Final expression for (6 \ imes \left(\frac{47}{11}\right)^{3/2}):\n [\n \frac{282\sqrt{517}}{121}\n ]", "---", "### Real-World Applyability", "These calculations emerge in physics, engineering, and data modeling where scaling factors, compound growth, or root transformations are involved. Understanding how to manipulate fractional exponents empowers better analysis of nonlinear relationships.", "---", "Conclusion\nBreaking down (x^3 = \left(\frac{47}{11}\right)^{3/2}) into simpler fractional exponents reveals (x = \left(\frac{47}{11}\right)^{1/2}), enabling precise computation of more complex expressions like (6 \ imes \left(\frac{47}{11}\right)^{3/2}), which simplifies elegantly to (\frac{282\sqrt{517}}{121}). Mastering these steps strengthens your algebraic toolkit and supports advanced problem-solving.", "---", "Keywords:\n(x^3 = \left(\frac{47}{11}\right)^{3/2}), how to calculate cube roots and fractional powers, simplify (\left(\frac{47}{11}\right)^{3/2}), compute (6 \ imes \left(\frac{47}{11}\right)^{3/2}), fractional exponent reduction, rationalizing radicals, algebra steps explained, exponential equations.", "---", "Meta Description:\nLearn step-by-step how to calculate (x^3 = \left(\frac{47}{11}\right)^{3/2}) and compute (6 \ imes \left(\frac{47}{11}\right)^{3/2}). Solve fractional exponents and simplify radicals with clear, practical examples.", "---", "See also:\n- Simplify fractional exponents\n- Rationalizing denominators with radicals\n- Solving exponential equations algebraically"]
