Solve {l}{sqrt{8}-sqrt{72}}{+sqrt{9/2}}

$$\left. \begin{array} { l } { \sqrt { 8 } - \sqrt { 72 } } \\ { + \sqrt { \frac { 9 } { 2 } } } \end{array} \right.$$

Evaluate

$$-\frac{5\sqrt{2}}{2}\approx -3.535533906$$

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}.

$$2\sqrt{2}-\sqrt{72}+\sqrt{\frac{9}{2}}$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$2\sqrt{2}-6\sqrt{2}+\sqrt{\frac{9}{2}}$$

Combine 2\sqrt{2} and -6\sqrt{2} to get -4\sqrt{2}.

$$-4\sqrt{2}+\sqrt{\frac{9}{2}}$$

Rewrite the square root of the division \sqrt{\frac{9}{2}} as the division of square roots \frac{\sqrt{9}}{\sqrt{2}}.

$$-4\sqrt{2}+\frac{\sqrt{9}}{\sqrt{2}}$$

Calculate the square root of 9 and get 3.

$$-4\sqrt{2}+\frac{3}{\sqrt{2}}$$

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

$$-4\sqrt{2}+\frac{3\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$-4\sqrt{2}+\frac{3\sqrt{2}}{2}$$

Combine -4\sqrt{2} and \frac{3\sqrt{2}}{2} to get -\frac{5}{2}\sqrt{2}.

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