\sqrt{\frac{10^2-2^6}{\frac{3^3-2^3}{2^2}}}-\left(-\frac{3}{7}\right)^2\left(-\frac{3}{7}\right)^{-3}\left(\frac{7}{3}\right)^{-2}

asked by guest
on Nov 25, 2024 at 6:44 pm



You asked:

Evaluate the expression: \(\sqrt{\frac{{10}^{2} - {2}^{6}}{\frac{{3}^{3} - {2}^{3}}{{2}^{2}}}} - {\left( -\left( \frac{3}{7} \right) \right)}^{2} {\left( -\left( \frac{3}{7} \right) \right)}^{-3} {\left( \frac{7}{3} \right)}^{-2}\)

MathBot Answer:

\[\sqrt{\frac{{10}^{2} - {2}^{6}}{\frac{{3}^{3} - {2}^{3}}{{2}^{2}}}} - {\left( -\left( \frac{3}{7} \right) \right)}^{2} {\left( -\left( \frac{3}{7} \right) \right)}^{-3} {\left( \frac{7}{3} \right)}^{-2} = \frac{3}{7} + \frac{12 \sqrt{19}}{19} \approx 3.18156023501816976231508636574592\]

\[\begin{aligned}\sqrt{\frac{{10}^{2} - {2}^{6}}{\frac{{3}^{3} - {2}^{3}}{{2}^{2}}}} - {\left( -\left( \frac{3}{7} \right) \right)}^{2} {\left( -\left( \frac{3}{7} \right) \right)}^{-3} {\left( \frac{7}{3} \right)}^{-2}& = \sqrt{\left(10^{2} - 2^{6}\right) \cdot \frac{1}{\left(3^{3} - 2^{3}\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{\left(100 - 2^{6}\right) \cdot \frac{1}{\left(3^{3} - 2^{3}\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{\left(100 - 64\right) \cdot \frac{1}{\left(3^{3} - 2^{3}\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{36 \cdot \frac{1}{\left(3^{3} - 2^{3}\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{36 \cdot \frac{1}{\left(27 - 2^{3}\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{36 \cdot \frac{1}{\left(27 - 8\right) \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{36 \cdot \frac{1}{19 \cdot \frac{1}{2^{2}}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{36 \cdot \frac{1}{\frac{19}{4}}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \sqrt{\frac{144}{19}} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \frac{12}{19} \cdot \sqrt{19} - \left(- \frac{3}{7}\right)^{2} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \frac{12}{19} \cdot \sqrt{19} - \frac{9}{49} \cdot \frac{1}{\left(- \frac{3}{7}\right)^{3}} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \frac{12}{19} \cdot \sqrt{19} - \frac{9}{49} \cdot - \frac{343}{27} \cdot \frac{1}{\left(\frac{7}{3}\right)^{2}}\\& = \frac{12}{19} \cdot \sqrt{19} - \frac{9}{49} \cdot - \frac{343}{27} \cdot \frac{9}{49}\\& = \frac{12}{19} \cdot \sqrt{19} - - \frac{7}{3} \cdot \frac{9}{49}\\& = \frac{12}{19} \cdot \sqrt{19} - - \frac{3}{7}\\& = \frac{12}{19} \cdot \sqrt{19} + \frac{3}{7}\\& = \frac{3}{7} + \frac{12}{19} \cdot \sqrt{19}\end{aligned}\]

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