Solve 1/5sqrt{180}+2sqrt{5}+3sqrt{170}+3sqrt{80}+3sqrt{5}

$$\frac{ 1 }{ 5 } \sqrt{ 180 } +2 \sqrt{ 5 } +3 \sqrt{ 170 } +3 \sqrt{ 80 } +3 \sqrt{ 5 }$$

Evaluate

$$\frac{91\sqrt{5}}{5}+3\sqrt{170}\approx 79.811651622$$

Solution Steps

Factor 180=6^{2}\times 5. Rewrite the square root of the product \sqrt{6^{2}\times 5} as the product of square roots \sqrt{6^{2}}\sqrt{5}. Take the square root of 6^{2}.

$$\frac{1}{5}\times 6\sqrt{5}+2\sqrt{5}+3\sqrt{170}+3\sqrt{80}+3\sqrt{5}$$

Multiply \frac{1}{5} and 6 to get \frac{6}{5}.

$$\frac{6}{5}\sqrt{5}+2\sqrt{5}+3\sqrt{170}+3\sqrt{80}+3\sqrt{5}$$

Combine \frac{6}{5}\sqrt{5} and 2\sqrt{5} to get \frac{16}{5}\sqrt{5}.

$$\frac{16}{5}\sqrt{5}+3\sqrt{170}+3\sqrt{80}+3\sqrt{5}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{16}{5}\sqrt{5}+3\sqrt{170}+3\times 4\sqrt{5}+3\sqrt{5}$$

Multiply 3 and 4 to get 12.

$$\frac{16}{5}\sqrt{5}+3\sqrt{170}+12\sqrt{5}+3\sqrt{5}$$

Combine \frac{16}{5}\sqrt{5} and 12\sqrt{5} to get \frac{76}{5}\sqrt{5}.

$$\frac{76}{5}\sqrt{5}+3\sqrt{170}+3\sqrt{5}$$

Combine \frac{76}{5}\sqrt{5} and 3\sqrt{5} to get \frac{91}{5}\sqrt{5}.

$$\frac{91}{5}\sqrt{5}+3\sqrt{170}$$

Factor

$$\frac{15 \sqrt{170} + 91 \sqrt{5}}{5} = 79.81165162171207$$