Solve sqrt{21/4}+3sqrt{28}*(-5sqrt{22/7})

$$\sqrt { 2 \frac { 1 } { 4 } } + 3 \sqrt { 28 } \times ( - 5 \sqrt { 2 \frac { 2 } { 7 } } )$$

Evaluate

$$-\frac{237}{2}=-118.5$$

Short Solution Steps

Multiply 2 and 4 to get 8.

$$\sqrt{\frac{8+1}{4}}+3\sqrt{28}\left(-5\right)\sqrt{\frac{2\times 7+2}{7}}$$

Add 8 and 1 to get 9.

$$\sqrt{\frac{9}{4}}+3\sqrt{28}\left(-5\right)\sqrt{\frac{2\times 7+2}{7}}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{3}{2}+3\sqrt{28}\left(-5\right)\sqrt{\frac{2\times 7+2}{7}}$$

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$\frac{3}{2}+3\times 2\sqrt{7}\left(-5\right)\sqrt{\frac{2\times 7+2}{7}}$$

Multiply 3 and 2 to get 6.

$$\frac{3}{2}+6\sqrt{7}\left(-5\right)\sqrt{\frac{2\times 7+2}{7}}$$

Multiply 6 and -5 to get -30.

$$\frac{3}{2}-30\sqrt{7}\sqrt{\frac{2\times 7+2}{7}}$$

Multiply 2 and 7 to get 14.

$$\frac{3}{2}-30\sqrt{7}\sqrt{\frac{14+2}{7}}$$

Add 14 and 2 to get 16.

$$\frac{3}{2}-30\sqrt{7}\sqrt{\frac{16}{7}}$$

Rewrite the square root of the division \sqrt{\frac{16}{7}} as the division of square roots \frac{\sqrt{16}}{\sqrt{7}}.

$$\frac{3}{2}-30\sqrt{7}\times \frac{\sqrt{16}}{\sqrt{7}}$$

Calculate the square root of 16 and get 4.

$$\frac{3}{2}-30\sqrt{7}\times \frac{4}{\sqrt{7}}$$

Rationalize the denominator of \frac{4}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{3}{2}-30\sqrt{7}\times \frac{4\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{3}{2}-30\sqrt{7}\times \frac{4\sqrt{7}}{7}$$

Express -30\times \frac{4\sqrt{7}}{7} as a single fraction.

$$\frac{3}{2}+\frac{-30\times 4\sqrt{7}}{7}\sqrt{7}$$

Multiply -30 and 4 to get -120.

$$\frac{3}{2}+\frac{-120\sqrt{7}}{7}\sqrt{7}$$

Express \frac{-120\sqrt{7}}{7}\sqrt{7} as a single fraction.

$$\frac{3}{2}+\frac{-120\sqrt{7}\sqrt{7}}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 7 is 14. Multiply \frac{3}{2} times \frac{7}{7}. Multiply \frac{-120\sqrt{7}\sqrt{7}}{7} times \frac{2}{2}.

$$\frac{3\times 7}{14}+\frac{2\left(-120\right)\sqrt{7}\sqrt{7}}{14}$$

Since \frac{3\times 7}{14} and \frac{2\left(-120\right)\sqrt{7}\sqrt{7}}{14} have the same denominator, add them by adding their numerators.

$$\frac{3\times 7+2\left(-120\right)\sqrt{7}\sqrt{7}}{14}$$

Do the multiplications in 3\times 7+2\left(-120\right)\sqrt{7}\sqrt{7}.

$$\frac{21-1680}{14}$$

Do the calculations in 21-1680.

$$\frac{-1659}{14}$$

Factor

$$-\frac{237}{2} = -118\frac{1}{2} = -118.5$$