Solve sqrt{45}-2sqrt{20}+frac{sqrt{405}}{3}+sqrt{80}

$$\sqrt { 45 } - 2 \sqrt { 20 } + \frac { \sqrt { 405 } } { 3 } + \sqrt { 80 }$$

Evaluate

$$6\sqrt{5}\approx 13.416407865$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$3\sqrt{5}-2\sqrt{20}+\frac{\sqrt{405}}{3}+\sqrt{80}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$3\sqrt{5}-2\times 2\sqrt{5}+\frac{\sqrt{405}}{3}+\sqrt{80}$$

Multiply -2 and 2 to get -4.

$$3\sqrt{5}-4\sqrt{5}+\frac{\sqrt{405}}{3}+\sqrt{80}$$

Combine 3\sqrt{5} and -4\sqrt{5} to get -\sqrt{5}.

$$-\sqrt{5}+\frac{\sqrt{405}}{3}+\sqrt{80}$$

Factor 405=9^{2}\times 5. Rewrite the square root of the product \sqrt{9^{2}\times 5} as the product of square roots \sqrt{9^{2}}\sqrt{5}. Take the square root of 9^{2}.

$$-\sqrt{5}+\frac{9\sqrt{5}}{3}+\sqrt{80}$$

Divide 9\sqrt{5} by 3 to get 3\sqrt{5}.

$$-\sqrt{5}+3\sqrt{5}+\sqrt{80}$$

Combine -\sqrt{5} and 3\sqrt{5} to get 2\sqrt{5}.

$$2\sqrt{5}+\sqrt{80}$$

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}.

$$2\sqrt{5}+4\sqrt{5}$$

Combine 2\sqrt{5} and 4\sqrt{5} to get 6\sqrt{5}.

$$6\sqrt{5}$$