Solve frac{sqrt{48}-sqrt{27}}{sqrt{3}}+sqrt{6}2sqrt{1/3}+sin(45)

$$\frac{ \sqrt{ 48 } - \sqrt{ 27 } }{ \sqrt{ 3 } } + \sqrt{ 6 } 2 \sqrt{ \frac{ 1 }{ 3 } } + \sin ( 45 )$$

Evaluate

$$\frac{5\sqrt{2}}{2}+1\approx 4.535533906$$

Short Solution Steps

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}.

$$\frac{4\sqrt{3}-\sqrt{27}}{\sqrt{3}}+2\sqrt{6}\sqrt{\frac{1}{3}}+\sin(45)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{4\sqrt{3}-3\sqrt{3}}{\sqrt{3}}+2\sqrt{6}\sqrt{\frac{1}{3}}+\sin(45)$$

Combine 4\sqrt{3} and -3\sqrt{3} to get \sqrt{3}.

$$\frac{\sqrt{3}}{\sqrt{3}}+2\sqrt{6}\sqrt{\frac{1}{3}}+\sin(45)$$

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

$$\sqrt{1}+2\sqrt{6}\sqrt{\frac{1}{3}}+\sin(45)$$

Calculate the square root of 1 and get 1.

$$1+2\sqrt{6}\sqrt{\frac{1}{3}}+\sin(45)$$

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

$$1+2\sqrt{6}\times \frac{\sqrt{1}}{\sqrt{3}}+\sin(45)$$

Calculate the square root of 1 and get 1.

$$1+2\sqrt{6}\times \frac{1}{\sqrt{3}}+\sin(45)$$

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

$$1+2\sqrt{6}\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sin(45)$$

The square of \sqrt{3} is 3.

$$1+2\sqrt{6}\times \frac{\sqrt{3}}{3}+\sin(45)$$

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

$$1+\frac{2\sqrt{3}}{3}\sqrt{6}+\sin(45)$$

Get the value of \sin(45) from trigonometric values table.

$$1+\frac{2\sqrt{3}}{3}\sqrt{6}+\frac{\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$\frac{2}{2}+\frac{2\sqrt{3}}{3}\sqrt{6}+\frac{\sqrt{2}}{2}$$

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

$$\frac{2+\sqrt{2}}{2}+\frac{2\sqrt{3}}{3}\sqrt{6}$$

Express \frac{2\sqrt{3}}{3}\sqrt{6} as a single fraction.

$$\frac{2+\sqrt{2}}{2}+\frac{2\sqrt{3}\sqrt{6}}{3}$$

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

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

Since \frac{3\left(2+\sqrt{2}\right)}{6} and \frac{2\times 2\sqrt{3}\sqrt{6}}{6} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(2+\sqrt{2}\right)+2\times 2\sqrt{3}\sqrt{6}.

$$\frac{6+3\sqrt{2}+12\sqrt{2}}{6}$$

Do the calculations in 6+3\sqrt{2}+12\sqrt{2}.

$$\frac{6+15\sqrt{2}}{6}$$