Solve sqrt{1000^2+(1000sqrt{5})^2}

$$\sqrt { 1000 ^ { 2 } + ( 1000 \sqrt { 5 } ) ^ { 2 } }$$

Evaluate

$$1000\sqrt{6}\approx 2449.489742783$$

Solution Steps

Calculate 1000 to the power of 2 and get 1000000.

$$\sqrt{1000000+\left(1000\sqrt{5}\right)^{2}}$$

Expand \left(1000\sqrt{5}\right)^{2}.

$$\sqrt{1000000+1000^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 1000 to the power of 2 and get 1000000.

$$\sqrt{1000000+1000000\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\sqrt{1000000+1000000\times 5}$$

Multiply 1000000 and 5 to get 5000000.

$$\sqrt{1000000+5000000}$$

Add 1000000 and 5000000 to get 6000000.

$$\sqrt{6000000}$$

Factor 6000000=1000^{2}\times 6. Rewrite the square root of the product \sqrt{1000^{2}\times 6} as the product of square roots \sqrt{1000^{2}}\sqrt{6}. Take the square root of 1000^{2}.

$$1000\sqrt{6}$$