Solve frac{(sqrt{6}+sqrt{10}*sqrt{15})*sqrt{3}}{(B)12sqrt{5}}

$$\frac { ( \sqrt { 6 } + \sqrt { 10 } \times \sqrt { 15 } ) \times \sqrt { 3 } } { ( B ) 12 \sqrt { 5 } }$$

Evaluate

$$\frac{3\sqrt{10}}{10B}$$

Short Solution Steps

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\left(\sqrt{6}+\sqrt{150}\right)\sqrt{3}}{12B\sqrt{5}}$$

Rationalize the denominator of \frac{\left(\sqrt{6}+\sqrt{150}\right)\sqrt{3}}{12B\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(\sqrt{6}+\sqrt{150}\right)\sqrt{3}\sqrt{5}}{12B\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(\sqrt{6}+\sqrt{150}\right)\sqrt{3}\sqrt{5}}{12B\times 5}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{\left(\sqrt{6}+5\sqrt{6}\right)\sqrt{3}\sqrt{5}}{12B\times 5}$$

Combine \sqrt{6} and 5\sqrt{6} to get 6\sqrt{6}.

$$\frac{6\sqrt{6}\sqrt{3}\sqrt{5}}{12B\times 5}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{6\sqrt{3}\sqrt{2}\sqrt{3}\sqrt{5}}{12B\times 5}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{6\times 3\sqrt{2}\sqrt{5}}{12B\times 5}$$

Multiply 6 and 3 to get 18.

$$\frac{18\sqrt{2}\sqrt{5}}{12B\times 5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{18\sqrt{10}}{12B\times 5}$$

Multiply 12 and 5 to get 60.

$$\frac{18\sqrt{10}}{60B}$$

Cancel out 6 in both numerator and denominator.

$$\frac{3\sqrt{10}}{10B}$$

Differentiate w.r.t. B

$$-\frac{3\sqrt{10}}{10B^{2}}$$