Solve sqrt{45}+3sqrt{20}+sqrt{80}+4sqrt{40}

$$\sqrt { 45 } + 3 \sqrt { 20 } + \sqrt { 80 } + 4 \sqrt { 40 }$$

Evaluate

$$8\sqrt{10}+13\sqrt{5}\approx 54.367104989$$

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}.

$$3\sqrt{5}+3\sqrt{20}+\sqrt{80}+4\sqrt{40}$$

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}.

$$3\sqrt{5}+3\times 2\sqrt{5}+\sqrt{80}+4\sqrt{40}$$

Multiply 3 and 2 to get 6.

$$3\sqrt{5}+6\sqrt{5}+\sqrt{80}+4\sqrt{40}$$

Combine 3\sqrt{5} and 6\sqrt{5} to get 9\sqrt{5}.

$$9\sqrt{5}+\sqrt{80}+4\sqrt{40}$$

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}.

$$9\sqrt{5}+4\sqrt{5}+4\sqrt{40}$$

Combine 9\sqrt{5} and 4\sqrt{5} to get 13\sqrt{5}.

$$13\sqrt{5}+4\sqrt{40}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$13\sqrt{5}+4\times 2\sqrt{10}$$

Multiply 4 and 2 to get 8.

$$13\sqrt{5}+8\sqrt{10}$$