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

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

Evaluate

$$\frac{-\sqrt{2}-7\sqrt{3}}{4}\approx -3.384642304$$

Solution Steps

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

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

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

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

Express -\frac{3}{4}\times 3 as a single fraction.

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

Multiply -3 and 3 to get -9.

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

Fraction \frac{-9}{4} can be rewritten as -\frac{9}{4} by extracting the negative sign.

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

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{7}{4}\sqrt{3}.

$$-\frac{1}{4}\sqrt{2}-\frac{7}{4}\sqrt{3}$$

Factor

$$\frac{-\sqrt{2} - 7 \sqrt{3}}{4} = -3.384642303838809$$