Solve sqrt{(2sqrt{78})^2-(5sqrt{6})^2}

$$\sqrt { ( 2 \sqrt { 78 } ) ^ { 2 } - ( 5 \sqrt { 6 } ) ^ { 2 } }$$

Evaluate

$$9\sqrt{2}\approx 12.727922061$$

Solution Steps

Expand \left(2\sqrt{78}\right)^{2}.

$$\sqrt{2^{2}\left(\sqrt{78}\right)^{2}-\left(5\sqrt{6}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{4\left(\sqrt{78}\right)^{2}-\left(5\sqrt{6}\right)^{2}}$$

The square of \sqrt{78} is 78.

$$\sqrt{4\times 78-\left(5\sqrt{6}\right)^{2}}$$

Multiply 4 and 78 to get 312.

$$\sqrt{312-\left(5\sqrt{6}\right)^{2}}$$

Expand \left(5\sqrt{6}\right)^{2}.

$$\sqrt{312-5^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{312-25\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{312-25\times 6}$$

Multiply 25 and 6 to get 150.

$$\sqrt{312-150}$$

Subtract 150 from 312 to get 162.

$$\sqrt{162}$$

Factor 162=9^{2}\times 2. Rewrite the square root of the product \sqrt{9^{2}\times 2} as the product of square roots \sqrt{9^{2}}\sqrt{2}. Take the square root of 9^{2}.

$$9\sqrt{2}$$