Solve sqrt{20^2+60^2}+100+sqrt{20^2+40^2}+sqrt{{40}^{2}+{80}^{2}}+sqrt{{20}^{2}+{60}^{2}}+sqrt{{100}^2+{40}^2}+sqrt{{40}^2+{50}^2}=

$$\sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } +100+ \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } + \sqrt{ { 100 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 50 }^{ 2 } } =$$

Evaluate

$$10\left(\sqrt{41}+2\sqrt{29}+4\sqrt{10}+6\sqrt{5}+10\right)\approx 532.389723574$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

$$\sqrt{400+60^{2}}+100+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 60 to the power of 2 and get 3600.

$$\sqrt{400+3600}+100+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Add 400 and 3600 to get 4000.

$$\sqrt{4000}+100+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Factor 4000=20^{2}\times 10. Rewrite the square root of the product \sqrt{20^{2}\times 10} as the product of square roots \sqrt{20^{2}}\sqrt{10}. Take the square root of 20^{2}.

$$20\sqrt{10}+100+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$20\sqrt{10}+100+\sqrt{400+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$20\sqrt{10}+100+\sqrt{400+1600}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Add 400 and 1600 to get 2000.

$$20\sqrt{10}+100+\sqrt{2000}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Factor 2000=20^{2}\times 5. Rewrite the square root of the product \sqrt{20^{2}\times 5} as the product of square roots \sqrt{20^{2}}\sqrt{5}. Take the square root of 20^{2}.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{1600+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 80 to the power of 2 and get 6400.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{1600+6400}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Add 1600 and 6400 to get 8000.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{8000}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Factor 8000=40^{2}\times 5. Rewrite the square root of the product \sqrt{40^{2}\times 5} as the product of square roots \sqrt{40^{2}}\sqrt{5}. Take the square root of 40^{2}.

$$20\sqrt{10}+100+20\sqrt{5}+40\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Combine 20\sqrt{5} and 40\sqrt{5} to get 60\sqrt{5}.

$$20\sqrt{10}+100+60\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$20\sqrt{10}+100+60\sqrt{5}+\sqrt{400+60^{2}}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 60 to the power of 2 and get 3600.

$$20\sqrt{10}+100+60\sqrt{5}+\sqrt{400+3600}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Add 400 and 3600 to get 4000.

$$20\sqrt{10}+100+60\sqrt{5}+\sqrt{4000}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Factor 4000=20^{2}\times 10. Rewrite the square root of the product \sqrt{20^{2}\times 10} as the product of square roots \sqrt{20^{2}}\sqrt{10}. Take the square root of 20^{2}.

$$20\sqrt{10}+100+60\sqrt{5}+20\sqrt{10}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Combine 20\sqrt{10} and 20\sqrt{10} to get 40\sqrt{10}.

$$40\sqrt{10}+100+60\sqrt{5}+\sqrt{100^{2}+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 100 to the power of 2 and get 10000.

$$40\sqrt{10}+100+60\sqrt{5}+\sqrt{10000+40^{2}}+\sqrt{40^{2}+50^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$40\sqrt{10}+100+60\sqrt{5}+\sqrt{10000+1600}+\sqrt{40^{2}+50^{2}}$$

Add 10000 and 1600 to get 11600.

$$40\sqrt{10}+100+60\sqrt{5}+\sqrt{11600}+\sqrt{40^{2}+50^{2}}$$

Factor 11600=20^{2}\times 29. Rewrite the square root of the product \sqrt{20^{2}\times 29} as the product of square roots \sqrt{20^{2}}\sqrt{29}. Take the square root of 20^{2}.

$$40\sqrt{10}+100+60\sqrt{5}+20\sqrt{29}+\sqrt{40^{2}+50^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$40\sqrt{10}+100+60\sqrt{5}+20\sqrt{29}+\sqrt{1600+50^{2}}$$

Calculate 50 to the power of 2 and get 2500.

$$40\sqrt{10}+100+60\sqrt{5}+20\sqrt{29}+\sqrt{1600+2500}$$

Add 1600 and 2500 to get 4100.

$$40\sqrt{10}+100+60\sqrt{5}+20\sqrt{29}+\sqrt{4100}$$

Factor 4100=10^{2}\times 41. Rewrite the square root of the product \sqrt{10^{2}\times 41} as the product of square roots \sqrt{10^{2}}\sqrt{41}. Take the square root of 10^{2}.

$$40\sqrt{10}+100+60\sqrt{5}+20\sqrt{29}+10\sqrt{41}$$

Factor

$$10 {(\sqrt{41} + 2 \sqrt{29} + 4 \sqrt{10} + 6 \sqrt{5} + 10)} = 532.389723574$$