Solve frac{27sqrt{2}+9sqrt{2}+9600sqrt{6}}{120sqrt{6}}

$$\frac{ 27 \sqrt{ 2 } +9 \sqrt{ 2 } +9600 \sqrt{ 6 } }{ 120 \sqrt{ 6 } }$$

Evaluate

$$\frac{\sqrt{3}}{10}+80\approx 80.173205081$$

Solution Steps

Combine 27\sqrt{2} and 9\sqrt{2} to get 36\sqrt{2}.

$$\frac{36\sqrt{2}+9600\sqrt{6}}{120\sqrt{6}}$$

Rationalize the denominator of \frac{36\sqrt{2}+9600\sqrt{6}}{120\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(36\sqrt{2}+9600\sqrt{6}\right)\sqrt{6}}{120\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(36\sqrt{2}+9600\sqrt{6}\right)\sqrt{6}}{120\times 6}$$

Multiply 120 and 6 to get 720.

$$\frac{\left(36\sqrt{2}+9600\sqrt{6}\right)\sqrt{6}}{720}$$

Use the distributive property to multiply 36\sqrt{2}+9600\sqrt{6} by \sqrt{6}.

$$\frac{36\sqrt{2}\sqrt{6}+9600\left(\sqrt{6}\right)^{2}}{720}$$

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}.

$$\frac{36\sqrt{2}\sqrt{2}\sqrt{3}+9600\left(\sqrt{6}\right)^{2}}{720}$$

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

$$\frac{36\times 2\sqrt{3}+9600\left(\sqrt{6}\right)^{2}}{720}$$

Multiply 36 and 2 to get 72.

$$\frac{72\sqrt{3}+9600\left(\sqrt{6}\right)^{2}}{720}$$

The square of \sqrt{6} is 6.

$$\frac{72\sqrt{3}+9600\times 6}{720}$$

Multiply 9600 and 6 to get 57600.

$$\frac{72\sqrt{3}+57600}{720}$$

Factor

$$\frac{\sqrt{3} + 800}{10} = 80.17320508075689$$