Solve frac{sqrt{20}+sqrt{5}}{sqrt{45}}-sqrt{1/3}*sqrt{6}

$$\frac { \sqrt { 20 } + \sqrt { 5 } } { \sqrt { 45 } } - \sqrt { \frac { 1 } { 3 } } \times \sqrt { 6 }$$

Evaluate

$$1-\sqrt{2}\approx -0.414213562$$

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}.

$$\frac{2\sqrt{5}+\sqrt{5}}{\sqrt{45}}-\sqrt{\frac{1}{3}}\sqrt{6}$$

Combine 2\sqrt{5} and \sqrt{5} to get 3\sqrt{5}.

$$\frac{3\sqrt{5}}{\sqrt{45}}-\sqrt{\frac{1}{3}}\sqrt{6}$$

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}.

$$\frac{3\sqrt{5}}{3\sqrt{5}}-\sqrt{\frac{1}{3}}\sqrt{6}$$

Cancel out 3\sqrt{5} in both numerator and denominator.

$$1-\sqrt{\frac{1}{3}}\sqrt{6}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$1-\frac{\sqrt{1}}{\sqrt{3}}\sqrt{6}$$

Calculate the square root of 1 and get 1.

$$1-\frac{1}{\sqrt{3}}\sqrt{6}$$

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

$$1-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{6}$$

The square of \sqrt{3} is 3.

$$1-\frac{\sqrt{3}}{3}\sqrt{6}$$

Express \frac{\sqrt{3}}{3}\sqrt{6} as a single fraction.

$$1-\frac{\sqrt{3}\sqrt{6}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$1-\frac{\sqrt{3}\sqrt{3}\sqrt{2}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$1-\frac{3\sqrt{2}}{3}$$

Cancel out 3 and 3.

$$1-\sqrt{2}$$