\(\sqrt{\frac{2\cdot 1.5\cdot 10^{-2}\cdot 1.6\cdot 10^{-27}}{1.6\cdot 10^{-19}\cdot 2\cdot 10^{4}}}\)

asked by guest
on Jan 04, 2025 at 10:57 pm



You asked:

Evaluate the expression: \(\sqrt{\frac{2 \cdot 1.5 \cdot {10}^{-2} \cdot 1.6 \cdot {10}^{-27}}{1.6 \cdot {10}^{-19} \cdot 2 \cdot {10}^{4}}}\)

MathBot Answer:

\[\sqrt{\frac{2 \cdot 1.5 \cdot {10}^{-2} \cdot 1.6 \cdot {10}^{-27}}{1.6 \cdot {10}^{-19} \cdot 2 \cdot {10}^{4}}} = \frac{\sqrt{6}}{20000000} \approx 0.0000001224744871391589049098642\]

\[\begin{aligned}\sqrt{\frac{2 \cdot 1.5 \cdot {10}^{-2} \cdot 1.6 \cdot {10}^{-27}}{1.6 \cdot {10}^{-19} \cdot 2 \cdot {10}^{4}}}& = \sqrt{2 \cdot \left(1 + \frac{5}{10}\right) \cdot \frac{1}{10^{2}} \cdot \left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \left(1 + \frac{1}{2}\right) \cdot \frac{1}{10^{2}} \cdot \left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{10^{2}} \cdot \left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \left(1 + \frac{3}{5}\right) \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{10^{27}} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\left(1 + \frac{6}{10}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\left(1 + \frac{3}{5}\right) \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{8}{5} \cdot \frac{1}{10^{19}} \cdot 2 \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{8}{5} \cdot \frac{2}{10000000000000000000} \cdot 10^{4}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{8}{5} \cdot \frac{2}{10000000000000000000} \cdot 10000}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{2}{6250000000000000000} \cdot 10000}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{10000}{3125000000000000000}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{1000000000000000000000000000} \cdot \frac{1}{\frac{1}{312500000000000}}}\\& = \sqrt{2 \cdot \frac{3}{2} \cdot \frac{1}{100} \cdot \frac{8}{5} \cdot \frac{1}{3200000000000}}\\& = \sqrt{\frac{3}{100} \cdot \frac{8}{5} \cdot \frac{1}{3200000000000}}\\& = \sqrt{\frac{6}{125} \cdot \frac{1}{3200000000000}}\\& = \sqrt{\frac{3}{200000000000000}}\\& = \frac{1}{20000000} \cdot \sqrt{6}\end{aligned}\]

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