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

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

Evaluate

$$37\sqrt{29}+57\sqrt{10}+80\sqrt{5}+100\approx 658.386362694$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

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

Calculate 60 to the power of 2 and get 3600.

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

Add 400 and 3600 to get 4000.

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

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}.

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

Calculate 20 to the power of 2 and get 400.

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

Calculate 40 to the power of 2 and get 1600.

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

Add 400 and 1600 to get 2000.

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

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}.

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

Calculate 20 to the power of 2 and get 400.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{400+40^{2}}+\sqrt{40^{2}+80^{2}}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{400+1600}+\sqrt{40^{2}+80^{2}}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

Add 400 and 1600 to get 2000.

$$20\sqrt{10}+100+20\sqrt{5}+\sqrt{2000}+\sqrt{40^{2}+80^{2}}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

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}.

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

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

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

Calculate 40 to the power of 2 and get 1600.

$$20\sqrt{10}+100+40\sqrt{5}+\sqrt{1600+80^{2}}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

Calculate 80 to the power of 2 and get 6400.

$$20\sqrt{10}+100+40\sqrt{5}+\sqrt{1600+6400}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

Add 1600 and 6400 to get 8000.

$$20\sqrt{10}+100+40\sqrt{5}+\sqrt{8000}+1.85\sqrt{20^{2}+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

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}.

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

Combine 40\sqrt{5} and 40\sqrt{5} to get 80\sqrt{5}.

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

Calculate 20 to the power of 2 and get 400.

$$20\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{400+60^{2}}+1.85\sqrt{40^{2}+100^{2}}$$

Calculate 60 to the power of 2 and get 3600.

$$20\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{400+3600}+1.85\sqrt{40^{2}+100^{2}}$$

Add 400 and 3600 to get 4000.

$$20\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{4000}+1.85\sqrt{40^{2}+100^{2}}$$

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}.

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

Multiply 1.85 and 20 to get 37.

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

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

$$57\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{40^{2}+100^{2}}$$

Calculate 40 to the power of 2 and get 1600.

$$57\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{1600+100^{2}}$$

Calculate 100 to the power of 2 and get 10000.

$$57\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{1600+10000}$$

Add 1600 and 10000 to get 11600.

$$57\sqrt{10}+100+80\sqrt{5}+1.85\sqrt{11600}$$

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}.

$$57\sqrt{10}+100+80\sqrt{5}+1.85\times 20\sqrt{29}$$

Multiply 1.85 and 20 to get 37.

$$57\sqrt{10}+100+80\sqrt{5}+37\sqrt{29}$$

Factor

$$37 \sqrt{29} + 57 \sqrt{10} + 80 \sqrt{5} + 100 = 658.386362694$$