Solve 1/4sqrt{80}-(5sqrt{1/18}+sqrt{45/4})

$$\frac { 1 } { 4 } \sqrt { 80 } - ( 5 \sqrt { \frac { 1 } { 18 } } + \sqrt { \frac { 45 } { 4 } } )$$

Evaluate

$$-\frac{\sqrt{5}}{2}-\frac{5\sqrt{2}}{6}\approx -2.296545291$$

Short Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{1}{4}\times 4\sqrt{5}-\left(5\sqrt{\frac{1}{18}}+\sqrt{\frac{45}{4}}\right)$$

Cancel out 4 and 4.

$$\sqrt{5}-\left(5\sqrt{\frac{1}{18}}+\sqrt{\frac{45}{4}}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{18}} as the division of square roots \frac{\sqrt{1}}{\sqrt{18}}.

$$\sqrt{5}-\left(5\times \frac{\sqrt{1}}{\sqrt{18}}+\sqrt{\frac{45}{4}}\right)$$

Calculate the square root of 1 and get 1.

$$\sqrt{5}-\left(5\times \frac{1}{\sqrt{18}}+\sqrt{\frac{45}{4}}\right)$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\sqrt{5}-\left(5\times \frac{1}{3\sqrt{2}}+\sqrt{\frac{45}{4}}\right)$$

Rationalize the denominator of \frac{1}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\sqrt{5}-\left(5\times \frac{\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}+\sqrt{\frac{45}{4}}\right)$$

The square of \sqrt{2} is 2.

$$\sqrt{5}-\left(5\times \frac{\sqrt{2}}{3\times 2}+\sqrt{\frac{45}{4}}\right)$$

Multiply 3 and 2 to get 6.

$$\sqrt{5}-\left(5\times \frac{\sqrt{2}}{6}+\sqrt{\frac{45}{4}}\right)$$

Express 5\times \frac{\sqrt{2}}{6} as a single fraction.

$$\sqrt{5}-\left(\frac{5\sqrt{2}}{6}+\sqrt{\frac{45}{4}}\right)$$

Rewrite the square root of the division \sqrt{\frac{45}{4}} as the division of square roots \frac{\sqrt{45}}{\sqrt{4}}.

$$\sqrt{5}-\left(\frac{5\sqrt{2}}{6}+\frac{\sqrt{45}}{\sqrt{4}}\right)$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\sqrt{5}-\left(\frac{5\sqrt{2}}{6}+\frac{3\sqrt{5}}{\sqrt{4}}\right)$$

Calculate the square root of 4 and get 2.

$$\sqrt{5}-\left(\frac{5\sqrt{2}}{6}+\frac{3\sqrt{5}}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 2 is 6. Multiply \frac{3\sqrt{5}}{2} times \frac{3}{3}.

$$\sqrt{5}-\left(\frac{5\sqrt{2}}{6}+\frac{3\times 3\sqrt{5}}{6}\right)$$

Since \frac{5\sqrt{2}}{6} and \frac{3\times 3\sqrt{5}}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{5}-\frac{5\sqrt{2}+3\times 3\sqrt{5}}{6}$$

Do the multiplications in 5\sqrt{2}+3\times 3\sqrt{5}.

$$\sqrt{5}-\frac{5\sqrt{2}+9\sqrt{5}}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{5} times \frac{6}{6}.

$$\frac{6\sqrt{5}}{6}-\frac{5\sqrt{2}+9\sqrt{5}}{6}$$

Since \frac{6\sqrt{5}}{6} and \frac{5\sqrt{2}+9\sqrt{5}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\sqrt{5}-\left(5\sqrt{2}+9\sqrt{5}\right)}{6}$$

Do the multiplications in 6\sqrt{5}-\left(5\sqrt{2}+9\sqrt{5}\right).

$$\frac{6\sqrt{5}-5\sqrt{2}-9\sqrt{5}}{6}$$

Do the calculations in 6\sqrt{5}-5\sqrt{2}-9\sqrt{5}.

$$\frac{-3\sqrt{5}-5\sqrt{2}}{6}$$

Factor

$$\frac{-3 \sqrt{5} - 5 \sqrt{2}}{6} = -2.296545290727474$$