Solve sqrt{332}+sqrt[3]{-27}+2sqrt{1/18}+1-sqrt{76})^2

$$\sqrt { 332 } + \sqrt[ 3 ] { - 27 } + 2 \sqrt { \frac { 1 } { 18 } } + 1 - \sqrt { 76 } ) ^ { 2 }$$

Evaluate

$$\frac{\sqrt{2}}{3}+2\sqrt{83}-2\sqrt{19}-2\approx 7.974473792$$

Short Solution Steps

Factor 332=2^{2}\times 83. Rewrite the square root of the product \sqrt{2^{2}\times 83} as the product of square roots \sqrt{2^{2}}\sqrt{83}. Take the square root of 2^{2}.

$$2\sqrt{83}+\sqrt[3]{-27}+2\sqrt{\frac{1}{18}}+1-\sqrt{76}$$

Calculate \sqrt[3]{-27} and get -3.

$$2\sqrt{83}-3+2\sqrt{\frac{1}{18}}+1-\sqrt{76}$$

Rewrite the square root of the division \sqrt{\frac{1}{18}} as the division of square roots \frac{\sqrt{1}}{\sqrt{18}}.

$$2\sqrt{83}-3+2\times \frac{\sqrt{1}}{\sqrt{18}}+1-\sqrt{76}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{83}-3+2\times \frac{1}{\sqrt{18}}+1-\sqrt{76}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$2\sqrt{83}-3+2\times \frac{1}{3\sqrt{2}}+1-\sqrt{76}$$

Rationalize the denominator of \frac{1}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\sqrt{83}-3+2\times \frac{\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}+1-\sqrt{76}$$

The square of \sqrt{2} is 2.

$$2\sqrt{83}-3+2\times \frac{\sqrt{2}}{3\times 2}+1-\sqrt{76}$$

Multiply 3 and 2 to get 6.

$$2\sqrt{83}-3+2\times \frac{\sqrt{2}}{6}+1-\sqrt{76}$$

Cancel out 6, the greatest common factor in 2 and 6.

$$2\sqrt{83}-3+\frac{\sqrt{2}}{3}+1-\sqrt{76}$$

Add -3 and 1 to get -2.

$$2\sqrt{83}-2+\frac{\sqrt{2}}{3}-\sqrt{76}$$

Factor 76=2^{2}\times 19. Rewrite the square root of the product \sqrt{2^{2}\times 19} as the product of square roots \sqrt{2^{2}}\sqrt{19}. Take the square root of 2^{2}.

$$2\sqrt{83}-2+\frac{\sqrt{2}}{3}-2\sqrt{19}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{83}-2-2\sqrt{19} times \frac{3}{3}.

$$\frac{3\left(2\sqrt{83}-2-2\sqrt{19}\right)}{3}+\frac{\sqrt{2}}{3}$$

Since \frac{3\left(2\sqrt{83}-2-2\sqrt{19}\right)}{3} and \frac{\sqrt{2}}{3} have the same denominator, add them by adding their numerators.

$$\frac{3\left(2\sqrt{83}-2-2\sqrt{19}\right)+\sqrt{2}}{3}$$

Do the multiplications in 3\left(2\sqrt{83}-2-2\sqrt{19}\right)+\sqrt{2}.

$$\frac{6\sqrt{83}-6-6\sqrt{19}+\sqrt{2}}{3}$$