Solve 1/3*3/sqrt{2}(36+16+sqrt{36+16})

$$\frac{ 1 }{ 3 } \times \frac{ 3 }{ \sqrt{ 2 } } (36+16+ \sqrt{ 36+16 } )$$

Evaluate

$$\sqrt{2}\left(\sqrt{13}+26\right)\approx 41.868572135$$

Solution Steps

Rationalize the denominator of \frac{3}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{1}{3}\times \frac{3\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\left(36+16+\sqrt{36+16}\right)$$

The square of \sqrt{2} is 2.

$$\frac{1}{3}\times \frac{3\sqrt{2}}{2}\left(36+16+\sqrt{36+16}\right)$$

Add 36 and 16 to get 52.

$$\frac{1}{3}\times \frac{3\sqrt{2}}{2}\left(52+\sqrt{36+16}\right)$$

Add 36 and 16 to get 52.

$$\frac{1}{3}\times \frac{3\sqrt{2}}{2}\left(52+\sqrt{52}\right)$$

Factor 52=2^{2}\times 13. Rewrite the square root of the product \sqrt{2^{2}\times 13} as the product of square roots \sqrt{2^{2}}\sqrt{13}. Take the square root of 2^{2}.

$$\frac{1}{3}\times \frac{3\sqrt{2}}{2}\left(52+2\sqrt{13}\right)$$

Multiply \frac{1}{3} times \frac{3\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{2}}{3\times 2}\left(52+2\sqrt{13}\right)$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{2}}{2}\left(52+2\sqrt{13}\right)$$

Express \frac{\sqrt{2}}{2}\left(52+2\sqrt{13}\right) as a single fraction.

$$\frac{\sqrt{2}\left(52+2\sqrt{13}\right)}{2}$$

Use the distributive property to multiply \sqrt{2} by 52+2\sqrt{13}.

$$\frac{52\sqrt{2}+2\sqrt{2}\sqrt{13}}{2}$$

To multiply \sqrt{2} and \sqrt{13}, multiply the numbers under the square root.

$$\frac{52\sqrt{2}+2\sqrt{26}}{2}$$

Divide each term of 52\sqrt{2}+2\sqrt{26} by 2 to get 26\sqrt{2}+\sqrt{26}.

$$26\sqrt{2}+\sqrt{26}$$

Factor

$$\sqrt{2} {(\sqrt{13} + 26)} = 41.868572135$$