Solve 1.85*sqrt{20^2+60^2}+1.85*sqrt{20^2+40^2}+sqrt{{20}^{2}+{40}^{2}}+sqrt{{40}^{2}+{80}^{2}}+sqrt{{20}^{2}+{60}^{2}}+sqrt{{20}^2+{80}^2}+sqrt{{40}^2+{100}^2}=

$$1.85 \times \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } +1.85 \times \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 100 }^{ 2 } } =$$

Evaluate

$$20\sqrt{17}+20\sqrt{29}+57\sqrt{10}+97\sqrt{5}\approx 587.313829102$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

$$1.85\sqrt{400+60^{2}}+1.85\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 60 to the power of 2 and get 3600.

$$1.85\sqrt{400+3600}+1.85\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Add 400 and 3600 to get 4000.

$$1.85\sqrt{4000}+1.85\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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}.

$$1.85\times 20\sqrt{10}+1.85\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Multiply 1.85 and 20 to get 37.

$$37\sqrt{10}+1.85\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$37\sqrt{10}+1.85\sqrt{400+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$37\sqrt{10}+1.85\sqrt{400+1600}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Add 400 and 1600 to get 2000.

$$37\sqrt{10}+1.85\sqrt{2000}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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}.

$$37\sqrt{10}+1.85\times 20\sqrt{5}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Multiply 1.85 and 20 to get 37.

$$37\sqrt{10}+37\sqrt{5}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$37\sqrt{10}+37\sqrt{5}+\sqrt{400+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$37\sqrt{10}+37\sqrt{5}+\sqrt{400+1600}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Add 400 and 1600 to get 2000.

$$37\sqrt{10}+37\sqrt{5}+\sqrt{2000}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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}.

$$37\sqrt{10}+37\sqrt{5}+20\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Combine 37\sqrt{5} and 20\sqrt{5} to get 57\sqrt{5}.

$$37\sqrt{10}+57\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$37\sqrt{10}+57\sqrt{5}+\sqrt{1600+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 80 to the power of 2 and get 6400.

$$37\sqrt{10}+57\sqrt{5}+\sqrt{1600+6400}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Add 1600 and 6400 to get 8000.

$$37\sqrt{10}+57\sqrt{5}+\sqrt{8000}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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}.

$$37\sqrt{10}+57\sqrt{5}+40\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Combine 57\sqrt{5} and 40\sqrt{5} to get 97\sqrt{5}.

$$37\sqrt{10}+97\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$37\sqrt{10}+97\sqrt{5}+\sqrt{400+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 60 to the power of 2 and get 3600.

$$37\sqrt{10}+97\sqrt{5}+\sqrt{400+3600}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Add 400 and 3600 to get 4000.

$$37\sqrt{10}+97\sqrt{5}+\sqrt{4000}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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}.

$$37\sqrt{10}+97\sqrt{5}+20\sqrt{10}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Combine 37\sqrt{10} and 20\sqrt{10} to get 57\sqrt{10}.

$$57\sqrt{10}+97\sqrt{5}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$57\sqrt{10}+97\sqrt{5}+\sqrt{400+80^{2}}+\sqrt{40^{2}+100^{2}}$$

Calculate 80 to the power of 2 and get 6400.

$$57\sqrt{10}+97\sqrt{5}+\sqrt{400+6400}+\sqrt{40^{2}+100^{2}}$$

Add 400 and 6400 to get 6800.

$$57\sqrt{10}+97\sqrt{5}+\sqrt{6800}+\sqrt{40^{2}+100^{2}}$$

Factor 6800=20^{2}\times 17. Rewrite the square root of the product \sqrt{20^{2}\times 17} as the product of square roots \sqrt{20^{2}}\sqrt{17}. Take the square root of 20^{2}.

$$57\sqrt{10}+97\sqrt{5}+20\sqrt{17}+\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$57\sqrt{10}+97\sqrt{5}+20\sqrt{17}+\sqrt{1600+100^{2}}$$

Calculate 100 to the power of 2 and get 10000.

$$57\sqrt{10}+97\sqrt{5}+20\sqrt{17}+\sqrt{1600+10000}$$

Add 1600 and 10000 to get 11600.

$$57\sqrt{10}+97\sqrt{5}+20\sqrt{17}+\sqrt{11600}$$

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}.

$$57\sqrt{10}+97\sqrt{5}+20\sqrt{17}+20\sqrt{29}$$

Factor

$$20 \sqrt{17} + 20 \sqrt{29} + 57 \sqrt{10} + 97 \sqrt{5} = 587.313829102$$