Solve frac{sqrt{8}asqrt{9}}{sqrt{6}}

$$\frac { \sqrt { 8 } a \sqrt { 9 } } { \sqrt { 6 } }$$

Evaluate

$$2\sqrt{3}a$$

Solution Steps

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}.

$$\frac{2\sqrt{2}a\sqrt{9}}{\sqrt{6}}$$

Calculate the square root of 9 and get 3.

$$\frac{2\sqrt{2}a\times 3}{\sqrt{6}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{2}a}{\sqrt{6}}$$

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

$$\frac{6\sqrt{2}a\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{6\sqrt{2}a\sqrt{6}}{6}$$

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

$$\frac{6\sqrt{2}a\sqrt{2}\sqrt{3}}{6}$$

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

$$\frac{6\times 2a\sqrt{3}}{6}$$

Cancel out 6 and 6.

$$2a\sqrt{3}$$

Differentiate w.r.t. a

$$2 \sqrt{3} = 3.464101615$$