Solve 4.5*10^-12*2.56*10^-2/2*8.85*10^-12*sqrt{{left(2.56*10^-2right)^2+{left(1.3*{10}^-2right)}^2}}

$$\frac{ 4.5 \times { 10 }^{ -12 } \times 2.56 \times { 10 }^{ -2 } }{ 2 \times 8.85 \times { 10 }^{ -12 } \times \sqrt{ { \left(2.56 \times { 10 }^{ -2 } \right) }^{ 2 } + { \left(1.3 \times { 10 }^{ -2 } \right) }^{ 2 } } }$$

Evaluate

$$\frac{1920\sqrt{20609}}{1215931}\approx 0.226683943$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -12 and -2 to get -14.

$$\frac{4.5\times 10^{-14}\times 2.56}{2\times 8.85\times 10^{-12}\sqrt{\left(2.56\times 10^{-2}\right)^{2}+\left(1.3\times 10^{-2}\right)^{2}}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{2.56\times 4.5}{2\times 8.85\times 10^{2}\sqrt{\left(1.3\times 10^{-2}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Multiply 2.56 and 4.5 to get 11.52.

$$\frac{11.52}{2\times 8.85\times 10^{2}\sqrt{\left(1.3\times 10^{-2}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Multiply 2 and 8.85 to get 17.7.

$$\frac{11.52}{17.7\times 10^{2}\sqrt{\left(1.3\times 10^{-2}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{11.52}{17.7\times 100\sqrt{\left(1.3\times 10^{-2}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Multiply 17.7 and 100 to get 1770.

$$\frac{11.52}{1770\sqrt{\left(1.3\times 10^{-2}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{11.52}{1770\sqrt{\left(1.3\times \frac{1}{100}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Multiply 1.3 and \frac{1}{100} to get \frac{13}{1000}.

$$\frac{11.52}{1770\sqrt{\left(\frac{13}{1000}\right)^{2}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Calculate \frac{13}{1000} to the power of 2 and get \frac{169}{1000000}.

$$\frac{11.52}{1770\sqrt{\frac{169}{1000000}+\left(2.56\times 10^{-2}\right)^{2}}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{11.52}{1770\sqrt{\frac{169}{1000000}+\left(2.56\times \frac{1}{100}\right)^{2}}}$$

Multiply 2.56 and \frac{1}{100} to get \frac{16}{625}.

$$\frac{11.52}{1770\sqrt{\frac{169}{1000000}+\left(\frac{16}{625}\right)^{2}}}$$

Calculate \frac{16}{625} to the power of 2 and get \frac{256}{390625}.

$$\frac{11.52}{1770\sqrt{\frac{169}{1000000}+\frac{256}{390625}}}$$

Add \frac{169}{1000000} and \frac{256}{390625} to get \frac{20609}{25000000}.

$$\frac{11.52}{1770\sqrt{\frac{20609}{25000000}}}$$

Rewrite the square root of the division \sqrt{\frac{20609}{25000000}} as the division of square roots \frac{\sqrt{20609}}{\sqrt{25000000}}.

$$\frac{11.52}{1770\times \frac{\sqrt{20609}}{\sqrt{25000000}}}$$

Calculate the square root of 25000000 and get 5000.

$$\frac{11.52}{1770\times \frac{\sqrt{20609}}{5000}}$$

Express 1770\times \frac{\sqrt{20609}}{5000} as a single fraction.

$$\frac{11.52}{\frac{1770\sqrt{20609}}{5000}}$$

Divide 11.52 by \frac{1770\sqrt{20609}}{5000} by multiplying 11.52 by the reciprocal of \frac{1770\sqrt{20609}}{5000}.

$$\frac{11.52\times 5000}{1770\sqrt{20609}}$$

Cancel out 10 in both numerator and denominator.

$$\frac{11.52\times 500}{177\sqrt{20609}}$$

Rationalize the denominator of \frac{11.52\times 500}{177\sqrt{20609}} by multiplying numerator and denominator by \sqrt{20609}.

$$\frac{11.52\times 500\sqrt{20609}}{177\left(\sqrt{20609}\right)^{2}}$$

The square of \sqrt{20609} is 20609.

$$\frac{11.52\times 500\sqrt{20609}}{177\times 20609}$$

Multiply 11.52 and 500 to get 5760.

$$\frac{5760\sqrt{20609}}{177\times 20609}$$

Multiply 177 and 20609 to get 3647793.

$$\frac{5760\sqrt{20609}}{3647793}$$

Divide 5760\sqrt{20609} by 3647793 to get \frac{1920}{1215931}\sqrt{20609}.

$$\frac{1920}{1215931}\sqrt{20609}$$