Solve sqrt{12}(3sqrt{50}-sqrt{162})-sqrt{18}(sqrt{432}-sqrt{192})

$$\sqrt { 12 } ( 3 \sqrt { 50 } - \sqrt { 162 } ) - \sqrt { 18 } ( \sqrt { 432 } - \sqrt { 192 } )$$

Evaluate

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Solution Steps

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}.

$$2\sqrt{3}\left(3\sqrt{50}-\sqrt{162}\right)-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$2\sqrt{3}\left(3\times 5\sqrt{2}-\sqrt{162}\right)-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

Multiply 3 and 5 to get 15.

$$2\sqrt{3}\left(15\sqrt{2}-\sqrt{162}\right)-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

Factor 162=9^{2}\times 2. Rewrite the square root of the product \sqrt{9^{2}\times 2} as the product of square roots \sqrt{9^{2}}\sqrt{2}. Take the square root of 9^{2}.

$$2\sqrt{3}\left(15\sqrt{2}-9\sqrt{2}\right)-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

Combine 15\sqrt{2} and -9\sqrt{2} to get 6\sqrt{2}.

$$2\sqrt{3}\times 6\sqrt{2}-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

Multiply 2 and 6 to get 12.

$$12\sqrt{3}\sqrt{2}-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

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

$$12\sqrt{6}-\sqrt{18}\left(\sqrt{432}-\sqrt{192}\right)$$

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}.

$$12\sqrt{6}-3\sqrt{2}\left(\sqrt{432}-\sqrt{192}\right)$$

Factor 432=12^{2}\times 3. Rewrite the square root of the product \sqrt{12^{2}\times 3} as the product of square roots \sqrt{12^{2}}\sqrt{3}. Take the square root of 12^{2}.

$$12\sqrt{6}-3\sqrt{2}\left(12\sqrt{3}-\sqrt{192}\right)$$

Factor 192=8^{2}\times 3. Rewrite the square root of the product \sqrt{8^{2}\times 3} as the product of square roots \sqrt{8^{2}}\sqrt{3}. Take the square root of 8^{2}.

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

Combine 12\sqrt{3} and -8\sqrt{3} to get 4\sqrt{3}.

$$12\sqrt{6}-3\sqrt{2}\times 4\sqrt{3}$$

Multiply 3 and 4 to get 12.

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

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

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

Combine 12\sqrt{6} and -12\sqrt{6} to get 0.

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Factor

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