Solve 2sqrt{363}+5sqrt{243}+sqrt{192}

$$2 \sqrt{ 363 } +5 \sqrt{ 243 } + \sqrt{ 192 }$$

Evaluate

$$75\sqrt{3}\approx 129.903810568$$

Solution Steps

Factor 363=11^{2}\times 3. Rewrite the square root of the product \sqrt{11^{2}\times 3} as the product of square roots \sqrt{11^{2}}\sqrt{3}. Take the square root of 11^{2}.

$$2\times 11\sqrt{3}+5\sqrt{243}+\sqrt{192}$$

Multiply 2 and 11 to get 22.

$$22\sqrt{3}+5\sqrt{243}+\sqrt{192}$$

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}.

$$22\sqrt{3}+5\times 9\sqrt{3}+\sqrt{192}$$

Multiply 5 and 9 to get 45.

$$22\sqrt{3}+45\sqrt{3}+\sqrt{192}$$

Combine 22\sqrt{3} and 45\sqrt{3} to get 67\sqrt{3}.

$$67\sqrt{3}+\sqrt{192}$$

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}.

$$67\sqrt{3}+8\sqrt{3}$$

Combine 67\sqrt{3} and 8\sqrt{3} to get 75\sqrt{3}.

$$75\sqrt{3}$$