Solve (sqrt{150}-sqrt{50})*(sqrt{48}+sqrt{108}-sqrt{27})sqrt{3}=

$$( \sqrt { 150 } - \sqrt { 50 } ) \cdot ( \sqrt { 48 } + \sqrt { 108 } - \sqrt { 27 } ) \sqrt { 3 } =$$

Evaluate

$$105\left(\sqrt{6}-\sqrt{2}\right)\approx 108.703998943$$

Solution Steps

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\left(5\sqrt{6}-\sqrt{50}\right)\left(\sqrt{48}+\sqrt{108}-\sqrt{27}\right)\sqrt{3}$$

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}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\left(\sqrt{48}+\sqrt{108}-\sqrt{27}\right)\sqrt{3}$$

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}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\left(4\sqrt{3}+\sqrt{108}-\sqrt{27}\right)\sqrt{3}$$

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}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\left(4\sqrt{3}+6\sqrt{3}-\sqrt{27}\right)\sqrt{3}$$

Combine 4\sqrt{3} and 6\sqrt{3} to get 10\sqrt{3}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\left(10\sqrt{3}-\sqrt{27}\right)\sqrt{3}$$

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}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\left(10\sqrt{3}-3\sqrt{3}\right)\sqrt{3}$$

Combine 10\sqrt{3} and -3\sqrt{3} to get 7\sqrt{3}.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\times 7\sqrt{3}\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\times 7\times 3$$

Multiply 7 and 3 to get 21.

$$\left(5\sqrt{6}-5\sqrt{2}\right)\times 21$$

Use the distributive property to multiply 5\sqrt{6}-5\sqrt{2} by 21.

$$105\sqrt{6}-105\sqrt{2}$$