Solve frac{sqrt{18}}{sqrt{3}}-(sqrt{3}+sqrt{2})^2

$$\frac { \sqrt { 18 } } { \sqrt { 3 } } - ( \sqrt { 3 } + \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$-\sqrt{6}-5\approx -7.449489743$$

Solution Steps

Rewrite the division of square roots \frac{\sqrt{18}}{\sqrt{3}} as the square root of the division \sqrt{\frac{18}{3}} and perform the division.

$$\sqrt{6}-\left(\sqrt{3}+\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+\sqrt{2}\right)^{2}.

$$\sqrt{6}-\left(\left(\sqrt{3}\right)^{2}+2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$\sqrt{6}-\left(3+2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\sqrt{6}-\left(3+2\sqrt{6}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

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

Add 3 and 2 to get 5.

$$\sqrt{6}-\left(5+2\sqrt{6}\right)$$

To find the opposite of 5+2\sqrt{6}, find the opposite of each term.

$$\sqrt{6}-5-2\sqrt{6}$$

Combine \sqrt{6} and -2\sqrt{6} to get -\sqrt{6}.

$$-\sqrt{6}-5$$