Solve sqrt{1/frac{1{149000000000^2}-1/298000000000^2}}

$$\sqrt{ \frac{ 1 }{ \frac{ 1 }{ { 149000000000 }^{ 2 } } - \frac{ 1 }{ { 298000000000 }^{ 2 } } } }$$

Evaluate

$$\frac{298000000000\sqrt{3}}{3}\approx 172050380218.508453369$$

Solution Steps

Calculate 149000000000 to the power of 2 and get 22201000000000000000000.

$$\sqrt{\frac{1}{\frac{1}{22201000000000000000000}-\frac{1}{298000000000^{2}}}}$$

Calculate 298000000000 to the power of 2 and get 88804000000000000000000.

$$\sqrt{\frac{1}{\frac{1}{22201000000000000000000}-\frac{1}{88804000000000000000000}}}$$

Least common multiple of 22201000000000000000000 and 88804000000000000000000 is 88804000000000000000000. Convert \frac{1}{22201000000000000000000} and \frac{1}{88804000000000000000000} to fractions with denominator 88804000000000000000000.

$$\sqrt{\frac{1}{\frac{4}{88804000000000000000000}-\frac{1}{88804000000000000000000}}}$$

Since \frac{4}{88804000000000000000000} and \frac{1}{88804000000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{\frac{4-1}{88804000000000000000000}}}$$

Subtract 1 from 4 to get 3.

$$\sqrt{\frac{1}{\frac{3}{88804000000000000000000}}}$$

Divide 1 by \frac{3}{88804000000000000000000} by multiplying 1 by the reciprocal of \frac{3}{88804000000000000000000}.

$$\sqrt{1\times \frac{88804000000000000000000}{3}}$$

Multiply 1 and \frac{88804000000000000000000}{3} to get \frac{88804000000000000000000}{3}.

$$\sqrt{\frac{88804000000000000000000}{3}}$$

Rewrite the square root of the division \sqrt{\frac{88804000000000000000000}{3}} as the division of square roots \frac{\sqrt{88804000000000000000000}}{\sqrt{3}}.

$$\frac{\sqrt{88804000000000000000000}}{\sqrt{3}}$$

Calculate the square root of 88804000000000000000000 and get 298000000000.

$$\frac{298000000000}{\sqrt{3}}$$

Rationalize the denominator of \frac{298000000000}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{298000000000\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{298000000000\sqrt{3}}{3}$$