Solve 1/2left(sqrt{2}+sqrt{3}right)-3/4left(sqrt{2}-sqrt{27}right)

$$\frac{ 1 }{ 2 } \left( \sqrt{ 2 } + \sqrt{ 3 } \right) - \frac{ 3 }{ 4 } \left( \sqrt{ 2 } - \sqrt{ 27 } \right)$$

Evaluate

$$\frac{11\sqrt{3}-\sqrt{2}}{4}\approx 4.40958633$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by \sqrt{2}+\sqrt{3}.

$$\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{3}-\frac{3}{4}\left(\sqrt{2}-\sqrt{27}\right)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{3}-\frac{3}{4}\left(\sqrt{2}-3\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{3}{4} by \sqrt{2}-3\sqrt{3}.

$$\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{3}-\frac{3}{4}\sqrt{2}-\frac{3}{4}\left(-3\right)\sqrt{3}$$

Express -\frac{3}{4}\left(-3\right) as a single fraction.

$$\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{3}-\frac{3}{4}\sqrt{2}+\frac{-3\left(-3\right)}{4}\sqrt{3}$$

Multiply -3 and -3 to get 9.

$$\frac{1}{2}\sqrt{2}+\frac{1}{2}\sqrt{3}-\frac{3}{4}\sqrt{2}+\frac{9}{4}\sqrt{3}$$

Combine \frac{1}{2}\sqrt{2} and -\frac{3}{4}\sqrt{2} to get -\frac{1}{4}\sqrt{2}.

$$-\frac{1}{4}\sqrt{2}+\frac{1}{2}\sqrt{3}+\frac{9}{4}\sqrt{3}$$

Combine \frac{1}{2}\sqrt{3} and \frac{9}{4}\sqrt{3} to get \frac{11}{4}\sqrt{3}.

$$-\frac{1}{4}\sqrt{2}+\frac{11}{4}\sqrt{3}$$

Factor

$$\frac{11 \sqrt{3} - \sqrt{2}}{4} = 4.409586330221138$$