Solve sqrt{(2000sqrt{6})^2+(3000sqrt{2})^2}

$$\sqrt { ( 2000 \sqrt { 6 } ) ^ { 2 } + ( 3000 \sqrt { 2 } ) ^ { 2 } }$$

Evaluate

$$1000\sqrt{42}\approx 6480.740698408$$

Solution Steps

Expand \left(2000\sqrt{6}\right)^{2}.

$$\sqrt{2000^{2}\left(\sqrt{6}\right)^{2}+\left(3000\sqrt{2}\right)^{2}}$$

Calculate 2000 to the power of 2 and get 4000000.

$$\sqrt{4000000\left(\sqrt{6}\right)^{2}+\left(3000\sqrt{2}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{4000000\times 6+\left(3000\sqrt{2}\right)^{2}}$$

Multiply 4000000 and 6 to get 24000000.

$$\sqrt{24000000+\left(3000\sqrt{2}\right)^{2}}$$

Expand \left(3000\sqrt{2}\right)^{2}.

$$\sqrt{24000000+3000^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3000 to the power of 2 and get 9000000.

$$\sqrt{24000000+9000000\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{24000000+9000000\times 2}$$

Multiply 9000000 and 2 to get 18000000.

$$\sqrt{24000000+18000000}$$

Add 24000000 and 18000000 to get 42000000.

$$\sqrt{42000000}$$

Factor 42000000=1000^{2}\times 42. Rewrite the square root of the product \sqrt{1000^{2}\times 42} as the product of square roots \sqrt{1000^{2}}\sqrt{42}. Take the square root of 1000^{2}.

$$1000\sqrt{42}$$