Solve sqrt{50}+sqrt{75}+sqrt{54}+sqrt{12}+sqrt{48}

$$\sqrt { 50 } + \sqrt { 75 } + \sqrt { 54 } + \sqrt { 12 } + \sqrt { 48 }$$

Evaluate

$$3\sqrt{6}+5\sqrt{2}+11\sqrt{3}\approx 33.472095923$$

Solution Steps

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}.

$$5\sqrt{2}+\sqrt{75}+\sqrt{54}+\sqrt{12}+\sqrt{48}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$5\sqrt{2}+5\sqrt{3}+\sqrt{54}+\sqrt{12}+\sqrt{48}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$5\sqrt{2}+5\sqrt{3}+3\sqrt{6}+\sqrt{12}+\sqrt{48}$$

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}.

$$5\sqrt{2}+5\sqrt{3}+3\sqrt{6}+2\sqrt{3}+\sqrt{48}$$

Combine 5\sqrt{3} and 2\sqrt{3} to get 7\sqrt{3}.

$$5\sqrt{2}+7\sqrt{3}+3\sqrt{6}+\sqrt{48}$$

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}.

$$5\sqrt{2}+7\sqrt{3}+3\sqrt{6}+4\sqrt{3}$$

Combine 7\sqrt{3} and 4\sqrt{3} to get 11\sqrt{3}.

$$5\sqrt{2}+11\sqrt{3}+3\sqrt{6}$$