Solve z^711*-120/20/sqrt{1000}

$$z ^ { 7 } \frac { 11 \cdot - 120 } { 20 / \sqrt { 1000 } }$$

Evaluate

$$-660\sqrt{10}z^{7}$$

Solution Steps

Multiply 11 and -120 to get -1320.

$$z^{7}\times \frac{-1320}{\frac{20}{\sqrt{1000}}}$$

Factor 1000=10^{2}\times 10. Rewrite the square root of the product \sqrt{10^{2}\times 10} as the product of square roots \sqrt{10^{2}}\sqrt{10}. Take the square root of 10^{2}.

$$z^{7}\times \frac{-1320}{\frac{20}{10\sqrt{10}}}$$

Rationalize the denominator of \frac{20}{10\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$z^{7}\times \frac{-1320}{\frac{20\sqrt{10}}{10\left(\sqrt{10}\right)^{2}}}$$

The square of \sqrt{10} is 10.

$$z^{7}\times \frac{-1320}{\frac{20\sqrt{10}}{10\times 10}}$$

Cancel out 2\times 10 in both numerator and denominator.

$$z^{7}\times \frac{-1320}{\frac{\sqrt{10}}{5}}$$

Divide -1320 by \frac{\sqrt{10}}{5} by multiplying -1320 by the reciprocal of \frac{\sqrt{10}}{5}.

$$z^{7}\times \frac{-1320\times 5}{\sqrt{10}}$$

Rationalize the denominator of \frac{-1320\times 5}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$z^{7}\times \frac{-1320\times 5\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$z^{7}\times \frac{-1320\times 5\sqrt{10}}{10}$$

Multiply -1320 and 5 to get -6600.

$$z^{7}\times \frac{-6600\sqrt{10}}{10}$$

Divide -6600\sqrt{10} by 10 to get -660\sqrt{10}.

$$z^{7}\left(-660\right)\sqrt{10}$$

Differentiate w.r.t. z

$$-4620\sqrt{10}z^{6}$$