Solve (3sqrt{6}+2sqrt{8}-sqrt{32})*sqrt{2}-sqrt{108}

$$(3 \sqrt{ 6 } +2 \sqrt{ 8 } - \sqrt{ 32 } ) \times \sqrt{ 2 } - \sqrt{ 108 }$$

Evaluate

$$0$$

Solution Steps

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}.

$$\left(3\sqrt{6}+2\times 2\sqrt{2}-\sqrt{32}\right)\sqrt{2}-\sqrt{108}$$

Multiply 2 and 2 to get 4.

$$\left(3\sqrt{6}+4\sqrt{2}-\sqrt{32}\right)\sqrt{2}-\sqrt{108}$$

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}.

$$\left(3\sqrt{6}+4\sqrt{2}-4\sqrt{2}\right)\sqrt{2}-\sqrt{108}$$

Combine 4\sqrt{2} and -4\sqrt{2} to get 0.

$$3\sqrt{6}\sqrt{2}-\sqrt{108}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$3\sqrt{2}\sqrt{3}\sqrt{2}-\sqrt{108}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$3\times 2\sqrt{3}-\sqrt{108}$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

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

Multiply 3 and 2 to get 6.

$$6\sqrt{3}-6\sqrt{3}$$

Combine 6\sqrt{3} and -6\sqrt{3} to get 0.

$$0$$

Factor

$$0$$