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

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

Evaluate

$$\frac{5\sqrt{2}+11\sqrt{3}}{4}\approx 6.530906674$$

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}.

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

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.

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

Combine \frac{1}{2}\sqrt{2} and \frac{3}{4}\sqrt{2} to get \frac{5}{4}\sqrt{2}.

$$\frac{5}{4}\sqrt{2}+\frac{1}{2}\sqrt{3}+\frac{9}{4}\sqrt{3}$$

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

$$\frac{5}{4}\sqrt{2}+\frac{11}{4}\sqrt{3}$$

Factor

$$\frac{5 \sqrt{2} + 11 \sqrt{3}}{4} = 6.530906673780781$$