Solve frac{sqrt{5}}{1-sqrt{1/23}}=

$$\frac { \sqrt { 5 } } { 1 - \sqrt { \frac { 1 } { 23 } } } =$$

Evaluate

$$\frac{\sqrt{115}\left(\sqrt{23}+1\right)}{22}\approx 2.825153126$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{23}} as the division of square roots \frac{\sqrt{1}}{\sqrt{23}}.

$$\frac{\sqrt{5}}{1-\frac{\sqrt{1}}{\sqrt{23}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{5}}{1-\frac{1}{\sqrt{23}}}$$

Rationalize the denominator of \frac{1}{\sqrt{23}} by multiplying numerator and denominator by \sqrt{23}.

$$\frac{\sqrt{5}}{1-\frac{\sqrt{23}}{\left(\sqrt{23}\right)^{2}}}$$

The square of \sqrt{23} is 23.

$$\frac{\sqrt{5}}{1-\frac{\sqrt{23}}{23}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{23}{23}.

$$\frac{\sqrt{5}}{\frac{23}{23}-\frac{\sqrt{23}}{23}}$$

Since \frac{23}{23} and \frac{\sqrt{23}}{23} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{5}}{\frac{23-\sqrt{23}}{23}}$$

Divide \sqrt{5} by \frac{23-\sqrt{23}}{23} by multiplying \sqrt{5} by the reciprocal of \frac{23-\sqrt{23}}{23}.

$$\frac{\sqrt{5}\times 23}{23-\sqrt{23}}$$

Rationalize the denominator of \frac{\sqrt{5}\times 23}{23-\sqrt{23}} by multiplying numerator and denominator by 23+\sqrt{23}.

$$\frac{\sqrt{5}\times 23\left(23+\sqrt{23}\right)}{\left(23-\sqrt{23}\right)\left(23+\sqrt{23}\right)}$$

Consider \left(23-\sqrt{23}\right)\left(23+\sqrt{23}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{5}\times 23\left(23+\sqrt{23}\right)}{23^{2}-\left(\sqrt{23}\right)^{2}}$$

Square 23. Square \sqrt{23}.

$$\frac{\sqrt{5}\times 23\left(23+\sqrt{23}\right)}{529-23}$$

Subtract 23 from 529 to get 506.

$$\frac{\sqrt{5}\times 23\left(23+\sqrt{23}\right)}{506}$$

Use the distributive property to multiply \sqrt{5}\times 23 by 23+\sqrt{23}.

$$\frac{23\sqrt{5}\times 23+23\sqrt{5}\sqrt{23}}{506}$$

Multiply 23 and 23 to get 529.

$$\frac{529\sqrt{5}+23\sqrt{5}\sqrt{23}}{506}$$

To multiply \sqrt{5} and \sqrt{23}, multiply the numbers under the square root.

$$\frac{529\sqrt{5}+23\sqrt{115}}{506}$$