Solve sqrt{128}-3sqrt{882}+2sqrt{162}+4sqrt{450}-sqrt{78}

$$\sqrt { 128 } - 3 \sqrt { 882 } + 2 \sqrt { 162 } + 4 \sqrt { 450 } - \sqrt { 78 }$$

Evaluate

$$23\sqrt{2}-\sqrt{78}\approx 23.695151068$$

Solution Steps

Factor 128=8^{2}\times 2. Rewrite the square root of the product \sqrt{8^{2}\times 2} as the product of square roots \sqrt{8^{2}}\sqrt{2}. Take the square root of 8^{2}.

$$8\sqrt{2}-3\sqrt{882}+2\sqrt{162}+4\sqrt{450}-\sqrt{78}$$

Factor 882=21^{2}\times 2. Rewrite the square root of the product \sqrt{21^{2}\times 2} as the product of square roots \sqrt{21^{2}}\sqrt{2}. Take the square root of 21^{2}.

$$8\sqrt{2}-3\times 21\sqrt{2}+2\sqrt{162}+4\sqrt{450}-\sqrt{78}$$

Multiply -3 and 21 to get -63.

$$8\sqrt{2}-63\sqrt{2}+2\sqrt{162}+4\sqrt{450}-\sqrt{78}$$

Combine 8\sqrt{2} and -63\sqrt{2} to get -55\sqrt{2}.

$$-55\sqrt{2}+2\sqrt{162}+4\sqrt{450}-\sqrt{78}$$

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}.

$$-55\sqrt{2}+2\times 9\sqrt{2}+4\sqrt{450}-\sqrt{78}$$

Multiply 2 and 9 to get 18.

$$-55\sqrt{2}+18\sqrt{2}+4\sqrt{450}-\sqrt{78}$$

Combine -55\sqrt{2} and 18\sqrt{2} to get -37\sqrt{2}.

$$-37\sqrt{2}+4\sqrt{450}-\sqrt{78}$$

Factor 450=15^{2}\times 2. Rewrite the square root of the product \sqrt{15^{2}\times 2} as the product of square roots \sqrt{15^{2}}\sqrt{2}. Take the square root of 15^{2}.

$$-37\sqrt{2}+4\times 15\sqrt{2}-\sqrt{78}$$

Multiply 4 and 15 to get 60.

$$-37\sqrt{2}+60\sqrt{2}-\sqrt{78}$$

Combine -37\sqrt{2} and 60\sqrt{2} to get 23\sqrt{2}.

$$23\sqrt{2}-\sqrt{78}$$