Solve 2sqrt{32}+4sqrt{128}-3sqrt{8}+6sqrt{16}+4sqrt{32}-2sqrt{512}

$$2 \sqrt { 32 } + 4 \sqrt { 128 } - 3 \sqrt { 8 } + 6 \sqrt { 16 } + 4 \sqrt { 32 } - 2 \sqrt { 512 }$$

Evaluate

$$18\sqrt{2}+24\approx 49.455844123$$

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}.

$$2\times 4\sqrt{2}+4\sqrt{128}-3\sqrt{8}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Multiply 2 and 4 to get 8.

$$8\sqrt{2}+4\sqrt{128}-3\sqrt{8}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

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}.

$$8\sqrt{2}+4\times 8\sqrt{2}-3\sqrt{8}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Multiply 4 and 8 to get 32.

$$8\sqrt{2}+32\sqrt{2}-3\sqrt{8}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Combine 8\sqrt{2} and 32\sqrt{2} to get 40\sqrt{2}.

$$40\sqrt{2}-3\sqrt{8}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

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

$$40\sqrt{2}-3\times 2\sqrt{2}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Multiply -3 and 2 to get -6.

$$40\sqrt{2}-6\sqrt{2}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Combine 40\sqrt{2} and -6\sqrt{2} to get 34\sqrt{2}.

$$34\sqrt{2}+6\sqrt{16}+4\sqrt{32}-2\sqrt{512}$$

Calculate the square root of 16 and get 4.

$$34\sqrt{2}+6\times 4+4\sqrt{32}-2\sqrt{512}$$

Multiply 6 and 4 to get 24.

$$34\sqrt{2}+24+4\sqrt{32}-2\sqrt{512}$$

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}.

$$34\sqrt{2}+24+4\times 4\sqrt{2}-2\sqrt{512}$$

Multiply 4 and 4 to get 16.

$$34\sqrt{2}+24+16\sqrt{2}-2\sqrt{512}$$

Combine 34\sqrt{2} and 16\sqrt{2} to get 50\sqrt{2}.

$$50\sqrt{2}+24-2\sqrt{512}$$

Factor 512=16^{2}\times 2. Rewrite the square root of the product \sqrt{16^{2}\times 2} as the product of square roots \sqrt{16^{2}}\sqrt{2}. Take the square root of 16^{2}.

$$50\sqrt{2}+24-2\times 16\sqrt{2}$$

Multiply -2 and 16 to get -32.

$$50\sqrt{2}+24-32\sqrt{2}$$

Combine 50\sqrt{2} and -32\sqrt{2} to get 18\sqrt{2}.

$$18\sqrt{2}+24$$