Solve frac{5*10^{-6}c*9*10^-6c}{4pi*1.0006*8.854*10^-12*c^2/Nm^2*1.5m}

$$\frac { 5 \cdot 10 ^ { - 6 } c \cdot 9 \cdot 10 ^ { - 6 } c } { 4 \pi \cdot 1.0006 \cdot 8.854 \cdot 10 ^ { - 12 } \cdot \frac { c ^ { 2 } } { N m ^ { 2 } } \cdot 1.5 m }$$

Evaluate

$$\frac{18750000Nm}{22148281\pi }$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{5\times 10^{-12}c\times 9c}{4\pi \times 1.0006\times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply c and c to get c^{2}.

$$\frac{5\times 10^{-12}c^{2}\times 9}{4\pi \times 1.0006\times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{5\times \frac{1}{1000000000000}c^{2}\times 9}{4\pi \times 1.0006\times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply 5 and \frac{1}{1000000000000} to get \frac{1}{200000000000}.

$$\frac{\frac{1}{200000000000}c^{2}\times 9}{4\pi \times 1.0006\times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply \frac{1}{200000000000} and 9 to get \frac{9}{200000000000}.

$$\frac{\frac{9}{200000000000}c^{2}}{4\pi \times 1.0006\times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply 4 and 1.0006 to get 4.0024.

$$\frac{\frac{9}{200000000000}c^{2}}{4.0024\pi \times 8.854\times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply 4.0024 and 8.854 to get 35.4372496.

$$\frac{\frac{9}{200000000000}c^{2}}{35.4372496\pi \times 10^{-12}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{\frac{9}{200000000000}c^{2}}{35.4372496\pi \times \frac{1}{1000000000000}\times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply 35.4372496 and \frac{1}{1000000000000} to get \frac{22148281}{625000000000000000}.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{22148281}{625000000000000000}\pi \times \frac{c^{2}}{Nm^{2}}\times 1.5m}$$

Multiply \frac{22148281}{625000000000000000} and 1.5 to get \frac{66444843}{1250000000000000000}.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{66444843}{1250000000000000000}\pi \times \frac{c^{2}}{Nm^{2}}m}$$

Multiply \frac{66444843}{1250000000000000000} times \frac{c^{2}}{Nm^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{66444843c^{2}}{1250000000000000000Nm^{2}}\pi m}$$

Express \frac{66444843c^{2}}{1250000000000000000Nm^{2}}m as a single fraction.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{66444843c^{2}m}{1250000000000000000Nm^{2}}\pi }$$

Cancel out m in both numerator and denominator.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{66444843c^{2}}{1250000000000000000Nm}\pi }$$

Express \frac{66444843c^{2}}{1250000000000000000Nm}\pi as a single fraction.

$$\frac{\frac{9}{200000000000}c^{2}}{\frac{66444843c^{2}\pi }{1250000000000000000Nm}}$$

Divide \frac{9}{200000000000}c^{2} by \frac{66444843c^{2}\pi }{1250000000000000000Nm} by multiplying \frac{9}{200000000000}c^{2} by the reciprocal of \frac{66444843c^{2}\pi }{1250000000000000000Nm}.

$$\frac{\frac{9}{200000000000}c^{2}\times 1250000000000000000Nm}{66444843c^{2}\pi }$$

Cancel out c^{2} in both numerator and denominator.

$$\frac{\frac{9}{200000000000}\times 1250000000000000000Nm}{66444843\pi }$$

Multiply \frac{9}{200000000000} and 1250000000000000000 to get 56250000.

$$\frac{56250000Nm}{66444843\pi }$$

Cancel out 3 in both numerator and denominator.

$$\frac{18750000Nm}{22148281\pi }$$

Differentiate w.r.t. N

$$\frac{18750000m}{22148281\pi }$$