Solve sqrt{80}-(sqrt{31/5}+5/4sqrt{45})

$$\sqrt { 80 } - ( \sqrt { 3 \frac { 1 } { 5 } } + \frac { 5 } { 4 } \sqrt { 45 } )$$

Evaluate

$$-\frac{11\sqrt{5}}{20}\approx -1.229837388$$

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}.

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

Multiply 3 and 5 to get 15.

$$4\sqrt{5}-\left(\sqrt{\frac{15+1}{5}}+\frac{5}{4}\sqrt{45}\right)$$

Add 15 and 1 to get 16.

$$4\sqrt{5}-\left(\sqrt{\frac{16}{5}}+\frac{5}{4}\sqrt{45}\right)$$

Rewrite the square root of the division \sqrt{\frac{16}{5}} as the division of square roots \frac{\sqrt{16}}{\sqrt{5}}.

$$4\sqrt{5}-\left(\frac{\sqrt{16}}{\sqrt{5}}+\frac{5}{4}\sqrt{45}\right)$$

Calculate the square root of 16 and get 4.

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

Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

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}.

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

Express \frac{5}{4}\times 3 as a single fraction.

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

Multiply 5 and 3 to get 15.

$$4\sqrt{5}-\left(\frac{4\sqrt{5}}{5}+\frac{15}{4}\sqrt{5}\right)$$

Combine \frac{4\sqrt{5}}{5} and \frac{15}{4}\sqrt{5} to get \frac{91}{20}\sqrt{5}.

$$4\sqrt{5}-\frac{91}{20}\sqrt{5}$$

Combine 4\sqrt{5} and -\frac{91}{20}\sqrt{5} to get -\frac{11}{20}\sqrt{5}.

$$-\frac{11}{20}\sqrt{5}$$