Cube Root Evaluation E87C85

1. **State the problem:** We need to evaluate the cube root of the expression $$\sqrt[3]{\frac{((24\times3600)^2 \times 6.67\times10^{-11} \times 5.97\times10^{24})}{4\pi^2}}.$$ 2. **Understand the formula:** The expression inside the cube root involves multiplication and division of constants and powers of ten. We will first calculate the numerator and denominator separately, then divide, and finally take the cube root. 3. **Calculate the numerator:** - Calculate $24 \times 3600$: $$24 \times 3600 = 86400.$$ - Square this value: $$86400^2 = 86400 \times 86400 = 7.46496 \times 10^9.$$ - Multiply by $6.67 \times 10^{-11}$: $$7.46496 \times 10^9 \times 6.67 \times 10^{-11} = (7.46496 \times 6.67) \times 10^{9 - 11} = 49.77 \times 10^{-2} = 0.4977.$$ - Multiply by $5.97 \times 10^{24}$: $$0.4977 \times 5.97 \times 10^{24} = (0.4977 \times 5.97) \times 10^{24} = 2.97 \times 10^{24}.$$ 4. **Calculate the denominator:** - Calculate $4\pi^2$: $$4 \times (3.1416)^2 = 4 \times 9.8696 = 39.4784.$$ 5. **Divide numerator by denominator:** $$\frac{2.97 \times 10^{24}}{39.4784} = 7.52 \times 10^{22}.$$ 6. **Take the cube root:** $$\sqrt[3]{7.52 \times 10^{22}} = \sqrt[3]{7.52} \times \sqrt[3]{10^{22}} = 1.96 \times 10^{\frac{22}{3}} = 1.96 \times 10^{7.33} = 1.96 \times 2.14 \times 10^{7} = 4.2 \times 10^{7}.$$ 7. **Final answer:** $$\boxed{4.2 \times 10^{7}}.$$ This is the value of the given cube root expression.