Solve sqrt{2}+3sqrt{32}+1/2sqrt{128}-6sqrt{18}

$$\sqrt { 2 } + 3 \sqrt { 32 } + \frac { 1 } { 2 } \sqrt { 128 } - 6 \sqrt { 18 }$$

Evaluate

$$-\sqrt{2}\approx -1.414213562$$

Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\sqrt{2}+3\times 4\sqrt{2}+\frac{1}{2}\sqrt{128}-6\sqrt{18}$$

Multiply 3 and 4 to get 12.

$$\sqrt{2}+12\sqrt{2}+\frac{1}{2}\sqrt{128}-6\sqrt{18}$$

Combine \sqrt{2} and 12\sqrt{2} to get 13\sqrt{2}.

$$13\sqrt{2}+\frac{1}{2}\sqrt{128}-6\sqrt{18}$$

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}.

$$13\sqrt{2}+\frac{1}{2}\times 8\sqrt{2}-6\sqrt{18}$$

Multiply \frac{1}{2} and 8 to get \frac{8}{2}.

$$13\sqrt{2}+\frac{8}{2}\sqrt{2}-6\sqrt{18}$$

Divide 8 by 2 to get 4.

$$13\sqrt{2}+4\sqrt{2}-6\sqrt{18}$$

Combine 13\sqrt{2} and 4\sqrt{2} to get 17\sqrt{2}.

$$17\sqrt{2}-6\sqrt{18}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$17\sqrt{2}-6\times 3\sqrt{2}$$

Multiply -6 and 3 to get -18.

$$17\sqrt{2}-18\sqrt{2}$$

Combine 17\sqrt{2} and -18\sqrt{2} to get -\sqrt{2}.

$$-\sqrt{2}$$