Solve sqrt{3}+sqrt{243}+sqrt{75}+sqrt{192}+sqrt{507}

$$\sqrt { 3 } + \sqrt { 243 } + \sqrt { 75 } + \sqrt { 192 } + \sqrt { 507 }$$

Evaluate

$$36\sqrt{3}\approx 62.353829072$$

Solution Steps

Factor 243=9^{2}\times 3. Rewrite the square root of the product \sqrt{9^{2}\times 3} as the product of square roots \sqrt{9^{2}}\sqrt{3}. Take the square root of 9^{2}.

$$\sqrt{3}+9\sqrt{3}+\sqrt{75}+\sqrt{192}+\sqrt{507}$$

Combine \sqrt{3} and 9\sqrt{3} to get 10\sqrt{3}.

$$10\sqrt{3}+\sqrt{75}+\sqrt{192}+\sqrt{507}$$

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}.

$$10\sqrt{3}+5\sqrt{3}+\sqrt{192}+\sqrt{507}$$

Combine 10\sqrt{3} and 5\sqrt{3} to get 15\sqrt{3}.

$$15\sqrt{3}+\sqrt{192}+\sqrt{507}$$

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}.

$$15\sqrt{3}+8\sqrt{3}+\sqrt{507}$$

Combine 15\sqrt{3} and 8\sqrt{3} to get 23\sqrt{3}.

$$23\sqrt{3}+\sqrt{507}$$

Factor 507=13^{2}\times 3. Rewrite the square root of the product \sqrt{13^{2}\times 3} as the product of square roots \sqrt{13^{2}}\sqrt{3}. Take the square root of 13^{2}.

$$23\sqrt{3}+13\sqrt{3}$$

Combine 23\sqrt{3} and 13\sqrt{3} to get 36\sqrt{3}.

$$36\sqrt{3}$$