Solve 3sqrt{27}+4sqrt{12}-3sqrt{48}+sqrt{192}

$$3 \sqrt { 27 } + 4 \sqrt { 12 } - 3 \sqrt { 48 } + \sqrt { 192 }$$

Evaluate

$$13\sqrt{3}\approx 22.516660498$$

Solution Steps

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}.

$$3\times 3\sqrt{3}+4\sqrt{12}-3\sqrt{48}+\sqrt{192}$$

Multiply 3 and 3 to get 9.

$$9\sqrt{3}+4\sqrt{12}-3\sqrt{48}+\sqrt{192}$$

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}.

$$9\sqrt{3}+4\times 2\sqrt{3}-3\sqrt{48}+\sqrt{192}$$

Multiply 4 and 2 to get 8.

$$9\sqrt{3}+8\sqrt{3}-3\sqrt{48}+\sqrt{192}$$

Combine 9\sqrt{3} and 8\sqrt{3} to get 17\sqrt{3}.

$$17\sqrt{3}-3\sqrt{48}+\sqrt{192}$$

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}.

$$17\sqrt{3}-3\times 4\sqrt{3}+\sqrt{192}$$

Multiply -3 and 4 to get -12.

$$17\sqrt{3}-12\sqrt{3}+\sqrt{192}$$

Combine 17\sqrt{3} and -12\sqrt{3} to get 5\sqrt{3}.

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

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}.

$$5\sqrt{3}+8\sqrt{3}$$

Combine 5\sqrt{3} and 8\sqrt{3} to get 13\sqrt{3}.

$$13\sqrt{3}$$