Solve 1+1/sqrt{2}*[frac{4+sqrt{256}}{sqrt{144}}]

$$1 + \frac { 1 } { \sqrt { 2 } } \times [ \frac { 4 + \sqrt { 256 } } { \sqrt { 144 } } ]$$

Evaluate

$$\frac{5\sqrt{2}}{6}+1\approx 2.178511302$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$1+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\times \frac{4+\sqrt{256}}{\sqrt{144}}$$

The square of \sqrt{2} is 2.

$$1+\frac{\sqrt{2}}{2}\times \frac{4+\sqrt{256}}{\sqrt{144}}$$

Calculate the square root of 256 and get 16.

$$1+\frac{\sqrt{2}}{2}\times \frac{4+16}{\sqrt{144}}$$

Add 4 and 16 to get 20.

$$1+\frac{\sqrt{2}}{2}\times \frac{20}{\sqrt{144}}$$

Calculate the square root of 144 and get 12.

$$1+\frac{\sqrt{2}}{2}\times \frac{20}{12}$$

Reduce the fraction \frac{20}{12} to lowest terms by extracting and canceling out 4.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2\times 3}{2\times 3}.

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

Since \frac{2\times 3}{2\times 3} and \frac{\sqrt{2}\times 5}{2\times 3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\times 3+\sqrt{2}\times 5.

$$\frac{6+5\sqrt{2}}{2\times 3}$$

Expand 2\times 3.

$$\frac{6+5\sqrt{2}}{6}$$