Solve 1+3sqrt{8}-1/3-1+sqrt{2}sin(45)

$$1+3 \sqrt{ 8 } - \frac{ 1 }{ 3 } -1+ \sqrt{ 2 } \sin ( 45 )$$

Evaluate

$$6\sqrt{2}+\frac{2}{3}\approx 9.151948041$$

Short Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

Multiply 3 and 2 to get 6.

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

Subtract \frac{1}{3} from 1 to get \frac{2}{3}.

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

Subtract 1 from \frac{2}{3} to get -\frac{1}{3}.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply -\frac{1}{3} times \frac{2}{2}. Multiply \frac{\sqrt{2}\sqrt{2}}{2} times \frac{3}{3}.

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

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

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

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

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

Do the calculations in -2+6.

$$\frac{4}{6}+6\sqrt{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6\sqrt{2} times \frac{6}{6}.

$$\frac{4}{6}+\frac{6\times 6\sqrt{2}}{6}$$

Since \frac{4}{6} and \frac{6\times 6\sqrt{2}}{6} have the same denominator, add them by adding their numerators.

$$\frac{4+6\times 6\sqrt{2}}{6}$$

Do the multiplications in 4+6\times 6\sqrt{2}.

$$\frac{4+36\sqrt{2}}{6}$$