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

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

Evaluate

$$\frac{-7\sqrt{2}-13\sqrt{3}}{4}\approx -8.104038859$$

Solution Steps

To find the opposite of \sqrt{2}+\sqrt{3}, find the opposite of each term.

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

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}.

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

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

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

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

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

Multiply -3 and 3 to get -9.

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

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

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

Combine -\sqrt{2} and -\frac{3}{4}\sqrt{2} to get -\frac{7}{4}\sqrt{2}.

$$-\frac{7}{4}\sqrt{2}-\sqrt{3}-\frac{9}{4}\sqrt{3}$$

Combine -\sqrt{3} and -\frac{9}{4}\sqrt{3} to get -\frac{13}{4}\sqrt{3}.

$$-\frac{7}{4}\sqrt{2}-\frac{13}{4}\sqrt{3}$$

Factor

$$\frac{-7 \sqrt{2} - 13 \sqrt{3}}{4} = -8.104038858751768$$