Solve frac{9sqrt{18}+9sqrt{6}+3200sqrt{54}}{240sqrt{6}}

$$\frac{ 9 \sqrt{ 18 } +9 \sqrt{ 6 } +3200 \sqrt{ 54 } }{ 240 \sqrt{ 6 } }$$

Evaluate

$$\frac{3\sqrt{3}+3203}{80}\approx 40.102451905$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{9\times 3\sqrt{2}+9\sqrt{6}+3200\sqrt{54}}{240\sqrt{6}}$$

Multiply 9 and 3 to get 27.

$$\frac{27\sqrt{2}+9\sqrt{6}+3200\sqrt{54}}{240\sqrt{6}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{27\sqrt{2}+9\sqrt{6}+3200\times 3\sqrt{6}}{240\sqrt{6}}$$

Multiply 3200 and 3 to get 9600.

$$\frac{27\sqrt{2}+9\sqrt{6}+9600\sqrt{6}}{240\sqrt{6}}$$

Combine 9\sqrt{6} and 9600\sqrt{6} to get 9609\sqrt{6}.

$$\frac{27\sqrt{2}+9609\sqrt{6}}{240\sqrt{6}}$$

Rationalize the denominator of \frac{27\sqrt{2}+9609\sqrt{6}}{240\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(27\sqrt{2}+9609\sqrt{6}\right)\sqrt{6}}{240\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(27\sqrt{2}+9609\sqrt{6}\right)\sqrt{6}}{240\times 6}$$

Multiply 240 and 6 to get 1440.

$$\frac{\left(27\sqrt{2}+9609\sqrt{6}\right)\sqrt{6}}{1440}$$

Use the distributive property to multiply 27\sqrt{2}+9609\sqrt{6} by \sqrt{6}.

$$\frac{27\sqrt{2}\sqrt{6}+9609\left(\sqrt{6}\right)^{2}}{1440}$$

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}.

$$\frac{27\sqrt{2}\sqrt{2}\sqrt{3}+9609\left(\sqrt{6}\right)^{2}}{1440}$$

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

$$\frac{27\times 2\sqrt{3}+9609\left(\sqrt{6}\right)^{2}}{1440}$$

Multiply 27 and 2 to get 54.

$$\frac{54\sqrt{3}+9609\left(\sqrt{6}\right)^{2}}{1440}$$

The square of \sqrt{6} is 6.

$$\frac{54\sqrt{3}+9609\times 6}{1440}$$

Multiply 9609 and 6 to get 57654.

$$\frac{54\sqrt{3}+57654}{1440}$$