Solve -4sqrt{27}+3sqrt{12}-(sqrt{48}+2sqrt{108}

$$- 4 \sqrt { 27 } + 3 \sqrt { 12 } - ( \sqrt { 48 } + 2 \sqrt { 108 }$$

Evaluate

$$-22\sqrt{3}\approx -38.105117767$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$-4\times 3\sqrt{3}+3\sqrt{12}-\left(\sqrt{48}+2\sqrt{108}\right)$$

Multiply -4 and 3 to get -12.

$$-12\sqrt{3}+3\sqrt{12}-\left(\sqrt{48}+2\sqrt{108}\right)$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$-12\sqrt{3}+3\times 2\sqrt{3}-\left(\sqrt{48}+2\sqrt{108}\right)$$

Multiply 3 and 2 to get 6.

$$-12\sqrt{3}+6\sqrt{3}-\left(\sqrt{48}+2\sqrt{108}\right)$$

Combine -12\sqrt{3} and 6\sqrt{3} to get -6\sqrt{3}.

$$-6\sqrt{3}-\left(\sqrt{48}+2\sqrt{108}\right)$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

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

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}.

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

Multiply 2 and 6 to get 12.

$$-6\sqrt{3}-\left(4\sqrt{3}+12\sqrt{3}\right)$$

Combine 4\sqrt{3} and 12\sqrt{3} to get 16\sqrt{3}.

$$-6\sqrt{3}-16\sqrt{3}$$

Combine -6\sqrt{3} and -16\sqrt{3} to get -22\sqrt{3}.

$$-22\sqrt{3}$$