Solve 1/5+sqrt{45}:quad(2)(sqrt{48}-sqrt{1/3})divsqrt{12}

$$\frac { 1 } { 5 } + \sqrt { 45 } : \quad ( 2 ) ( \sqrt { 48 } - \sqrt { \frac { 1 } { 3 } } ) \div \sqrt { 12 }$$

Evaluate

$$\frac{11\sqrt{5}}{4}+\frac{1}{5}\approx 6.349186938$$

Short Solution Steps

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}.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(\sqrt{48}-\sqrt{\frac{1}{3}}\right)}{\sqrt{12}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(4\sqrt{3}-\sqrt{\frac{1}{3}}\right)}{\sqrt{12}}$$

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

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(4\sqrt{3}-\frac{\sqrt{1}}{\sqrt{3}}\right)}{\sqrt{12}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(4\sqrt{3}-\frac{1}{\sqrt{3}}\right)}{\sqrt{12}}$$

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

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(4\sqrt{3}-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)}{\sqrt{12}}$$

The square of \sqrt{3} is 3.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(4\sqrt{3}-\frac{\sqrt{3}}{3}\right)}{\sqrt{12}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{3} times \frac{3}{3}.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\left(\frac{3\times 4\sqrt{3}}{3}-\frac{\sqrt{3}}{3}\right)}{\sqrt{12}}$$

Since \frac{3\times 4\sqrt{3}}{3} and \frac{\sqrt{3}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\times \frac{3\times 4\sqrt{3}-\sqrt{3}}{3}}{\sqrt{12}}$$

Do the multiplications in 3\times 4\sqrt{3}-\sqrt{3}.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\times \frac{12\sqrt{3}-\sqrt{3}}{3}}{\sqrt{12}}$$

Do the calculations in 12\sqrt{3}-\sqrt{3}.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}}{2}\times \frac{11\sqrt{3}}{3}}{\sqrt{12}}$$

Multiply \frac{3\sqrt{5}}{2} times \frac{11\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{5}+\frac{\frac{3\sqrt{5}\times 11\sqrt{3}}{2\times 3}}{\sqrt{12}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{1}{5}+\frac{\frac{11\sqrt{3}\sqrt{5}}{2}}{\sqrt{12}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{1}{5}+\frac{\frac{11\sqrt{3}\sqrt{5}}{2}}{2\sqrt{3}}$$

Express \frac{\frac{11\sqrt{3}\sqrt{5}}{2}}{2\sqrt{3}} as a single fraction.

$$\frac{1}{5}+\frac{11\sqrt{3}\sqrt{5}}{2\times 2\sqrt{3}}$$

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

$$\frac{1}{5}+\frac{11\sqrt{5}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{1}{5}+\frac{11\sqrt{5}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 4 is 20. Multiply \frac{1}{5} times \frac{4}{4}. Multiply \frac{11\sqrt{5}}{4} times \frac{5}{5}.

$$\frac{4}{20}+\frac{5\times 11\sqrt{5}}{20}$$

Since \frac{4}{20} and \frac{5\times 11\sqrt{5}}{20} have the same denominator, add them by adding their numerators.

$$\frac{4+5\times 11\sqrt{5}}{20}$$

Do the multiplications in 4+5\times 11\sqrt{5}.

$$\frac{4+55\sqrt{5}}{20}$$

Factor

$$\frac{55 \sqrt{5} + 4}{20} = 6.349186938124422$$