Solve 2*frac{sqrt{7}a}{5}*frac{sqrt{30}}{6sqrt{2}}

$$2 \cdot \frac { \sqrt { 7 } a } { 5 } \cdot \frac { \sqrt { 30 } } { 6 \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{105}a}{15}$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{30}}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{\sqrt{30}\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{\sqrt{30}\sqrt{2}}{6\times 2}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{\sqrt{2}\sqrt{15}\sqrt{2}}{6\times 2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{2\sqrt{15}}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{2\sqrt{15}}{12}$$

Divide 2\sqrt{15} by 12 to get \frac{1}{6}\sqrt{15}.

$$2\times \frac{\sqrt{7}a}{5}\times \frac{1}{6}\sqrt{15}$$

Multiply 2 and \frac{1}{6} to get \frac{2}{6}.

$$\frac{2}{6}\times \frac{\sqrt{7}a}{5}\sqrt{15}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}\times \frac{\sqrt{7}a}{5}\sqrt{15}$$

Multiply \frac{1}{3} times \frac{\sqrt{7}a}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{7}a}{3\times 5}\sqrt{15}$$

Express \frac{\sqrt{7}a}{3\times 5}\sqrt{15} as a single fraction.

$$\frac{\sqrt{7}a\sqrt{15}}{3\times 5}$$

To multiply \sqrt{7} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\sqrt{105}a}{3\times 5}$$

Multiply 3 and 5 to get 15.

$$\frac{\sqrt{105}a}{15}$$

Differentiate w.r.t. a

$$\frac{\sqrt{105}}{15} = 0.6831300510639732$$