Solve frac{sqrt{14}}{8}*3/4+frac{3sqrt{8}}{8}*frac{sqrt{7}}{4}

$$\frac { \sqrt { 14 } } { 8 } \times \frac { 3 } { 4 } + \frac { 3 \sqrt { 8 } } { 8 } \times \frac { \sqrt { 7 } } { 4 }$$

Evaluate

$$\frac{9\sqrt{14}}{32}\approx 1.05234114$$

Short Solution Steps

Multiply \frac{\sqrt{14}}{8} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{14}\times 3}{8\times 4}+\frac{3\sqrt{8}}{8}\times \frac{\sqrt{7}}{4}$$

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}.

$$\frac{\sqrt{14}\times 3}{8\times 4}+\frac{3\times 2\sqrt{2}}{8}\times \frac{\sqrt{7}}{4}$$

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{14}\times 3}{8\times 4}+\frac{6\sqrt{2}}{8}\times \frac{\sqrt{7}}{4}$$

Divide 6\sqrt{2} by 8 to get \frac{3}{4}\sqrt{2}.

$$\frac{\sqrt{14}\times 3}{8\times 4}+\frac{3}{4}\sqrt{2}\times \frac{\sqrt{7}}{4}$$

Multiply \frac{3}{4} times \frac{\sqrt{7}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{14}\times 3}{8\times 4}+\frac{3\sqrt{7}}{4\times 4}\sqrt{2}$$

Multiply 8 and 4 to get 32.

$$\frac{\sqrt{14}\times 3}{32}+\frac{3\sqrt{7}}{4\times 4}\sqrt{2}$$

Multiply 4 and 4 to get 16.

$$\frac{\sqrt{14}\times 3}{32}+\frac{3\sqrt{7}}{16}\sqrt{2}$$

Express \frac{3\sqrt{7}}{16}\sqrt{2} as a single fraction.

$$\frac{\sqrt{14}\times 3}{32}+\frac{3\sqrt{7}\sqrt{2}}{16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 32 and 16 is 32. Multiply \frac{3\sqrt{7}\sqrt{2}}{16} times \frac{2}{2}.

$$\frac{\sqrt{14}\times 3}{32}+\frac{2\times 3\sqrt{7}\sqrt{2}}{32}$$

Since \frac{\sqrt{14}\times 3}{32} and \frac{2\times 3\sqrt{7}\sqrt{2}}{32} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{14}\times 3+2\times 3\sqrt{7}\sqrt{2}}{32}$$

Do the multiplications in \sqrt{14}\times 3+2\times 3\sqrt{7}\sqrt{2}.

$$\frac{3\sqrt{14}+6\sqrt{14}}{32}$$

Do the calculations in 3\sqrt{14}+6\sqrt{14}.

$$\frac{9\sqrt{14}}{32}$$