Solve sqrt{18}(sqrt{108}-2sqrt{48})-sqrt{12}(sqrt{288}-sqrt{72})

$$\sqrt { 18 } ( \sqrt { 108 } - 2 \sqrt { 48 } ) - \sqrt { 12 } ( \sqrt { 288 } - \sqrt { 72 } )$$

Evaluate

$$-18\sqrt{6}\approx -44.09081537$$

Solution Steps

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}.

$$3\sqrt{2}\left(\sqrt{108}-2\sqrt{48}\right)-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

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}.

$$3\sqrt{2}\left(6\sqrt{3}-2\sqrt{48}\right)-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

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}.

$$3\sqrt{2}\left(6\sqrt{3}-2\times 4\sqrt{3}\right)-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

Multiply -2 and 4 to get -8.

$$3\sqrt{2}\left(6\sqrt{3}-8\sqrt{3}\right)-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

Combine 6\sqrt{3} and -8\sqrt{3} to get -2\sqrt{3}.

$$3\sqrt{2}\left(-2\right)\sqrt{3}-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

Multiply 3 and -2 to get -6.

$$-6\sqrt{2}\sqrt{3}-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$-6\sqrt{6}-\sqrt{12}\left(\sqrt{288}-\sqrt{72}\right)$$

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}.

$$-6\sqrt{6}-2\sqrt{3}\left(\sqrt{288}-\sqrt{72}\right)$$

Factor 288=12^{2}\times 2. Rewrite the square root of the product \sqrt{12^{2}\times 2} as the product of square roots \sqrt{12^{2}}\sqrt{2}. Take the square root of 12^{2}.

$$-6\sqrt{6}-2\sqrt{3}\left(12\sqrt{2}-\sqrt{72}\right)$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

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

Combine 12\sqrt{2} and -6\sqrt{2} to get 6\sqrt{2}.

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

Multiply 2 and 6 to get 12.

$$-6\sqrt{6}-12\sqrt{3}\sqrt{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$-6\sqrt{6}-12\sqrt{6}$$

Combine -6\sqrt{6} and -12\sqrt{6} to get -18\sqrt{6}.

$$-18\sqrt{6}$$