Solve 1/a_{1sqrt{20}}*sqrt{5}-frac{sqrt{75}}{sqrt{3}}

$$\frac { 1 } { a _ { 1 } \sqrt { 20 } } \cdot \sqrt { 5 } - \frac { \sqrt { 75 } } { \sqrt { 3 } }$$

Evaluate

$$-5+\frac{1}{2a_{1}}$$

Short Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{1}{a_{1}\times 2\sqrt{5}}\sqrt{5}-\frac{\sqrt{75}}{\sqrt{3}}$$

Rationalize the denominator of \frac{1}{a_{1}\times 2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{5}}{a_{1}\times 2\left(\sqrt{5}\right)^{2}}\sqrt{5}-\frac{\sqrt{75}}{\sqrt{3}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}}{a_{1}\times 2\times 5}\sqrt{5}-\frac{\sqrt{75}}{\sqrt{3}}$$

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{5}}{a_{1}\times 10}\sqrt{5}-\frac{\sqrt{75}}{\sqrt{3}}$$

Express \frac{\sqrt{5}}{a_{1}\times 10}\sqrt{5} as a single fraction.

$$\frac{\sqrt{5}\sqrt{5}}{a_{1}\times 10}-\frac{\sqrt{75}}{\sqrt{3}}$$

Rewrite the division of square roots \frac{\sqrt{75}}{\sqrt{3}} as the square root of the division \sqrt{\frac{75}{3}} and perform the division.

$$\frac{\sqrt{5}\sqrt{5}}{a_{1}\times 10}-\sqrt{25}$$

Calculate the square root of 25 and get 5.

$$\frac{\sqrt{5}\sqrt{5}}{a_{1}\times 10}-5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{a_{1}\times 10}{a_{1}\times 10}.

$$\frac{\sqrt{5}\sqrt{5}}{a_{1}\times 10}-\frac{5a_{1}\times 10}{a_{1}\times 10}$$

Since \frac{\sqrt{5}\sqrt{5}}{a_{1}\times 10} and \frac{5a_{1}\times 10}{a_{1}\times 10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{5}\sqrt{5}-5a_{1}\times 10}{a_{1}\times 10}$$

Do the multiplications in \sqrt{5}\sqrt{5}-5a_{1}\times 10.

$$\frac{5-50a_{1}}{a_{1}\times 10}$$

Factor the expressions that are not already factored in \frac{5-50a_{1}}{a_{1}\times 10}.

$$\frac{5\left(-10a_{1}+1\right)}{10a_{1}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{-10a_{1}+1}{2a_{1}}$$

Factor

$$-\frac{\frac{1}{2}\left(10a_{1}-1\right)}{a_{1}}$$