Solve frac{sqrt{frac{1+40.8*1.6*10^{-19}}{2*1.67*10^{-27}*(3*10^{8})^2}}{2*1.67*10^-27*(3*10^8)^2}}{2}

$$\frac { \sqrt { \frac { 1 + 40.8 \times 1.6 \times 10 ^ { - 19 } } { 2 \times 1.67 \times 10 ^ { - 27 } \times ( 3 \times 10 ^ { 8 } ) ^ { 2 } } } { 2 \times 1.67 \times 10 ^ { - 27 } \times ( 3 \times 10 ^ { 8 } ) ^ { 2 } } } { 2 }$$

Evaluate

$$\frac{3\sqrt{1304687500000000008517}}{12500000000000000}\approx 0.000008669$$

Solution Steps

Cancel out 2 and 2.

$$\sqrt{\frac{1+40.8\times 1.6\times 10^{-19}}{2\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 40.8 and 1.6 to get 65.28.

$$\sqrt{\frac{1+65.28\times 10^{-19}}{2\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 10 to the power of -19 and get \frac{1}{10000000000000000000}.

$$\sqrt{\frac{1+65.28\times \frac{1}{10000000000000000000}}{2\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 65.28 and \frac{1}{10000000000000000000} to get \frac{51}{7812500000000000000}.

$$\sqrt{\frac{1+\frac{51}{7812500000000000000}}{2\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Add 1 and \frac{51}{7812500000000000000} to get \frac{7812500000000000051}{7812500000000000000}.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{2\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 2 and 1.67 to get 3.34.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{3.34\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 10 to the power of -27 and get \frac{1}{1000000000000000000000000000}.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{3.34\times \frac{1}{1000000000000000000000000000}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 3.34 and \frac{1}{1000000000000000000000000000} to get \frac{167}{50000000000000000000000000000}.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{\frac{167}{50000000000000000000000000000}\times \left(3\times 10^{8}\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 10 to the power of 8 and get 100000000.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{\frac{167}{50000000000000000000000000000}\times \left(3\times 100000000\right)^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 3 and 100000000 to get 300000000.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{\frac{167}{50000000000000000000000000000}\times 300000000^{2}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 300000000 to the power of 2 and get 90000000000000000.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{\frac{167}{50000000000000000000000000000}\times 90000000000000000}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply \frac{167}{50000000000000000000000000000} and 90000000000000000 to get \frac{1503}{5000000000000}.

$$\sqrt{\frac{\frac{7812500000000000051}{7812500000000000000}}{\frac{1503}{5000000000000}}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Divide \frac{7812500000000000051}{7812500000000000000} by \frac{1503}{5000000000000} by multiplying \frac{7812500000000000051}{7812500000000000000} by the reciprocal of \frac{1503}{5000000000000}.

$$\sqrt{\frac{7812500000000000051}{7812500000000000000}\times \frac{5000000000000}{1503}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply \frac{7812500000000000051}{7812500000000000000} and \frac{5000000000000}{1503} to get \frac{7812500000000000051}{2348437500}.

$$\sqrt{\frac{7812500000000000051}{2348437500}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Rewrite the square root of the division \sqrt{\frac{7812500000000000051}{2348437500}} as the division of square roots \frac{\sqrt{7812500000000000051}}{\sqrt{2348437500}}.

$$\frac{\sqrt{7812500000000000051}}{\sqrt{2348437500}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Factor 2348437500=3750^{2}\times 167. Rewrite the square root of the product \sqrt{3750^{2}\times 167} as the product of square roots \sqrt{3750^{2}}\sqrt{167}. Take the square root of 3750^{2}.

$$\frac{\sqrt{7812500000000000051}}{3750\sqrt{167}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Rationalize the denominator of \frac{\sqrt{7812500000000000051}}{3750\sqrt{167}} by multiplying numerator and denominator by \sqrt{167}.

$$\frac{\sqrt{7812500000000000051}\sqrt{167}}{3750\left(\sqrt{167}\right)^{2}}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

The square of \sqrt{167} is 167.

$$\frac{\sqrt{7812500000000000051}\sqrt{167}}{3750\times 167}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

To multiply \sqrt{7812500000000000051} and \sqrt{167}, multiply the numbers under the square root.

$$\frac{\sqrt{1304687500000000008517}}{3750\times 167}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 3750 and 167 to get 626250.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times 1.67\times 10^{-27}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 10 to the power of -27 and get \frac{1}{1000000000000000000000000000}.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times 1.67\times \frac{1}{1000000000000000000000000000}\times \left(3\times 10^{8}\right)^{2}$$

Multiply 1.67 and \frac{1}{1000000000000000000000000000} to get \frac{167}{100000000000000000000000000000}.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times \frac{167}{100000000000000000000000000000}\times \left(3\times 10^{8}\right)^{2}$$

Calculate 10 to the power of 8 and get 100000000.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times \frac{167}{100000000000000000000000000000}\times \left(3\times 100000000\right)^{2}$$

Multiply 3 and 100000000 to get 300000000.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times \frac{167}{100000000000000000000000000000}\times 300000000^{2}$$

Calculate 300000000 to the power of 2 and get 90000000000000000.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times \frac{167}{100000000000000000000000000000}\times 90000000000000000$$

Multiply \frac{167}{100000000000000000000000000000} and 90000000000000000 to get \frac{1503}{10000000000000}.

$$\frac{\sqrt{1304687500000000008517}}{626250}\times \frac{1503}{10000000000000}$$

Multiply \frac{\sqrt{1304687500000000008517}}{626250} times \frac{1503}{10000000000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{1304687500000000008517}\times 1503}{626250\times 10000000000000}$$

Cancel out 501 in both numerator and denominator.

$$\frac{3\sqrt{1304687500000000008517}}{1250\times 10000000000000}$$

Multiply 1250 and 10000000000000 to get 12500000000000000.

$$\frac{3\sqrt{1304687500000000008517}}{12500000000000000}$$