Solve 1/2sqrt{3}-3/2sqrt{2}+1/4sqrt{8}-2/sqrt{3}

$$\frac { 1 } { 2 } \sqrt { 3 } - \frac { 3 } { 2 \sqrt { 2 } } + \frac { 1 } { 4 } \sqrt { 8 } - \frac { 2 } { \sqrt { 3 } }$$

Evaluate

$$-\frac{\sqrt{2}}{4}-\frac{\sqrt{3}}{6}\approx -0.642228525$$

Solution Steps

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

$$\frac{1}{2}\sqrt{3}-\frac{3\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\frac{1}{4}\sqrt{8}-\frac{2}{\sqrt{3}}$$

The square of \sqrt{2} is 2.

$$\frac{1}{2}\sqrt{3}-\frac{3\sqrt{2}}{2\times 2}+\frac{1}{4}\sqrt{8}-\frac{2}{\sqrt{3}}$$

Multiply 2 and 2 to get 4.

$$\frac{1}{2}\sqrt{3}-\frac{3\sqrt{2}}{4}+\frac{1}{4}\sqrt{8}-\frac{2}{\sqrt{3}}$$

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}.

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

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Combine \frac{1}{2}\sqrt{3} and -\frac{2\sqrt{3}}{3} to get -\frac{1}{6}\sqrt{3}.

$$-\frac{1}{6}\sqrt{3}-\frac{1}{4}\sqrt{2}$$

Factor

$$\frac{-2 \sqrt{3} - 3 \sqrt{2}}{12} = -0.6422285251880867$$