Solve 1/sqrt{frac{1{20^2}+(1/30-1/180)^2}}

$$\frac { 1 } { \sqrt { \frac { 1 } { 20 ^ { 2 } } + ( \frac { 1 } { 30 } - \frac { 1 } { 180 } ) ^ { 2 } } }$$

Evaluate

$$\frac{90\sqrt{106}}{53}\approx 17.483145522$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

$$\frac{1}{\sqrt{\frac{1}{400}+\left(\frac{1}{30}-\frac{1}{180}\right)^{2}}}$$

Subtract \frac{1}{180} from \frac{1}{30} to get \frac{1}{36}.

$$\frac{1}{\sqrt{\frac{1}{400}+\left(\frac{1}{36}\right)^{2}}}$$

Calculate \frac{1}{36} to the power of 2 and get \frac{1}{1296}.

$$\frac{1}{\sqrt{\frac{1}{400}+\frac{1}{1296}}}$$

Add \frac{1}{400} and \frac{1}{1296} to get \frac{53}{16200}.

$$\frac{1}{\sqrt{\frac{53}{16200}}}$$

Rewrite the square root of the division \sqrt{\frac{53}{16200}} as the division of square roots \frac{\sqrt{53}}{\sqrt{16200}}.

$$\frac{1}{\frac{\sqrt{53}}{\sqrt{16200}}}$$

Factor 16200=90^{2}\times 2. Rewrite the square root of the product \sqrt{90^{2}\times 2} as the product of square roots \sqrt{90^{2}}\sqrt{2}. Take the square root of 90^{2}.

$$\frac{1}{\frac{\sqrt{53}}{90\sqrt{2}}}$$

Rationalize the denominator of \frac{\sqrt{53}}{90\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{1}{\frac{\sqrt{53}\sqrt{2}}{90\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{1}{\frac{\sqrt{53}\sqrt{2}}{90\times 2}}$$

To multiply \sqrt{53} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{1}{\frac{\sqrt{106}}{90\times 2}}$$

Multiply 90 and 2 to get 180.

$$\frac{1}{\frac{\sqrt{106}}{180}}$$

Divide 1 by \frac{\sqrt{106}}{180} by multiplying 1 by the reciprocal of \frac{\sqrt{106}}{180}.

$$\frac{180}{\sqrt{106}}$$

Rationalize the denominator of \frac{180}{\sqrt{106}} by multiplying numerator and denominator by \sqrt{106}.

$$\frac{180\sqrt{106}}{\left(\sqrt{106}\right)^{2}}$$

The square of \sqrt{106} is 106.

$$\frac{180\sqrt{106}}{106}$$

Divide 180\sqrt{106} by 106 to get \frac{90}{53}\sqrt{106}.

$$\frac{90}{53}\sqrt{106}$$