Solve [(14)(1.66*10^{-27})]=[frac{(1.602*10^-19)(0.5)^2(R_{0})^2}{2(2.26*10^4)}]

$$[ ( 14 ) ( 1.66 \times 10 ^ { - 27 } ) ] = [ \frac { ( 1.602 \times 10 ^ { - 19 } ) ( 0.5 ) ^ { 2 } ( R _ { 0 } ) ^ { 2 } } { 2 ( 2.26 \times 10 ^ { 4 } ) } ]$$

Solve for R_0

$$R_{0}=\frac{\sqrt{29215585}}{33375}\approx 0.161951889$$

$$R_{0}=-\frac{\sqrt{29215585}}{33375}\approx -0.161951889$$

Steps by Finding Square Root

Multiply 14 and 1.66 to get 23.24.

$$23.24\times 10^{-27}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

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

$$23.24\times \frac{1}{1000000000000000000000000000}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

Multiply 23.24 and \frac{1}{1000000000000000000000000000} to get \frac{581}{25000000000000000000000000000}.

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

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

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 0.25R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Multiply 1.602 and 0.25 to get 0.4005.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Multiply 2 and 2.26 to get 4.52.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{4.52\times 10^{23}}$$

Calculate 10 to the power of 23 and get 100000000000000000000000.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{4.52\times 100000000000000000000000}$$

Multiply 4.52 and 100000000000000000000000 to get 452000000000000000000000.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{452000000000000000000000}$$

Divide 0.4005R_{0}^{2} by 452000000000000000000000 to get \frac{801}{904000000000000000000000000}R_{0}^{2}.

$$\frac{581}{25000000000000000000000000000}=\frac{801}{904000000000000000000000000}R_{0}^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{801}{904000000000000000000000000}R_{0}^{2}=\frac{581}{25000000000000000000000000000}$$

Divide both sides by \frac{801}{904000000000000000000000000}.

$$R_{0}^{2}=\frac{\frac{581}{25000000000000000000000000000}}{\frac{801}{904000000000000000000000000}}$$

Express \frac{\frac{581}{25000000000000000000000000000}}{\frac{801}{904000000000000000000000000}} as a single fraction.

$$R_{0}^{2}=\frac{581}{25000000000000000000000000000\times \frac{801}{904000000000000000000000000}}$$

Multiply 25000000000000000000000000000 and \frac{801}{904000000000000000000000000} to get \frac{2503125}{113}.

$$R_{0}^{2}=\frac{581}{\frac{2503125}{113}}$$

Take the square root of both sides of the equation.

$$R_{0}=\frac{\sqrt{29215585}}{33375} R_{0}=-\frac{\sqrt{29215585}}{33375}$$

Steps Using the Quadratic Formula

Multiply 14 and 1.66 to get 23.24.

$$23.24\times 10^{-27}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

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

$$23.24\times \frac{1}{1000000000000000000000000000}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

Multiply 23.24 and \frac{1}{1000000000000000000000000000} to get \frac{581}{25000000000000000000000000000}.

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 10^{-19}\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{4}}$$

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

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 0.5^{2}R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\frac{581}{25000000000000000000000000000}=\frac{1.602\times 0.25R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Multiply 1.602 and 0.25 to get 0.4005.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{2\times 2.26\times 10^{23}}$$

Multiply 2 and 2.26 to get 4.52.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{4.52\times 10^{23}}$$

Calculate 10 to the power of 23 and get 100000000000000000000000.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{4.52\times 100000000000000000000000}$$

Multiply 4.52 and 100000000000000000000000 to get 452000000000000000000000.

$$\frac{581}{25000000000000000000000000000}=\frac{0.4005R_{0}^{2}}{452000000000000000000000}$$

Divide 0.4005R_{0}^{2} by 452000000000000000000000 to get \frac{801}{904000000000000000000000000}R_{0}^{2}.

$$\frac{581}{25000000000000000000000000000}=\frac{801}{904000000000000000000000000}R_{0}^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{801}{904000000000000000000000000}R_{0}^{2}=\frac{581}{25000000000000000000000000000}$$

Subtract \frac{581}{25000000000000000000000000000} from both sides.

$$\frac{801}{904000000000000000000000000}R_{0}^{2}-\frac{581}{25000000000000000000000000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{801}{904000000000000000000000000} for a, 0 for b, and -\frac{581}{25000000000000000000000000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$R_{0}=\frac{0±\sqrt{0^{2}-4\times \frac{801}{904000000000000000000000000}\left(-\frac{581}{25000000000000000000000000000}\right)}}{2\times \frac{801}{904000000000000000000000000}}$$

Square 0.

$$R_{0}=\frac{0±\sqrt{-4\times \frac{801}{904000000000000000000000000}\left(-\frac{581}{25000000000000000000000000000}\right)}}{2\times \frac{801}{904000000000000000000000000}}$$

Multiply -4 times \frac{801}{904000000000000000000000000}.

$$R_{0}=\frac{0±\sqrt{-\frac{801}{226000000000000000000000000}\left(-\frac{581}{25000000000000000000000000000}\right)}}{2\times \frac{801}{904000000000000000000000000}}$$

Multiply -\frac{801}{226000000000000000000000000} times -\frac{581}{25000000000000000000000000000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$R_{0}=\frac{0±\sqrt{\frac{465381}{5650000000000000000000000000000000000000000000000000000}}}{2\times \frac{801}{904000000000000000000000000}}$$

Take the square root of \frac{465381}{5650000000000000000000000000000000000000000000000000000}.

$$R_{0}=\frac{0±\frac{3\sqrt{29215585}}{56500000000000000000000000000}}{2\times \frac{801}{904000000000000000000000000}}$$

Multiply 2 times \frac{801}{904000000000000000000000000}.

$$R_{0}=\frac{0±\frac{3\sqrt{29215585}}{56500000000000000000000000000}}{\frac{801}{452000000000000000000000000}}$$

Now solve the equation R_{0}=\frac{0±\frac{3\sqrt{29215585}}{56500000000000000000000000000}}{\frac{801}{452000000000000000000000000}} when ± is plus.

$$R_{0}=\frac{\sqrt{29215585}}{33375}$$

Now solve the equation R_{0}=\frac{0±\frac{3\sqrt{29215585}}{56500000000000000000000000000}}{\frac{801}{452000000000000000000000000}} when ± is minus.

$$R_{0}=-\frac{\sqrt{29215585}}{33375}$$

The equation is now solved.

$$R_{0}=\frac{\sqrt{29215585}}{33375} R_{0}=-\frac{\sqrt{29215585}}{33375}$$