Solve 3sqrt{27}+5sqrt{78}+sqrt{76}-2sqrt{12}

$$3 \sqrt { 27 } + 5 \sqrt { 78 } + \sqrt { 76 } - 2 \sqrt { 12 }$$

Evaluate

$$2\sqrt{19}+5\sqrt{3}+5\sqrt{78}\approx 61.536856257$$

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}.

$$3\times 3\sqrt{3}+5\sqrt{78}+\sqrt{76}-2\sqrt{12}$$

Multiply 3 and 3 to get 9.

$$9\sqrt{3}+5\sqrt{78}+\sqrt{76}-2\sqrt{12}$$

Factor 76=2^{2}\times 19. Rewrite the square root of the product \sqrt{2^{2}\times 19} as the product of square roots \sqrt{2^{2}}\sqrt{19}. Take the square root of 2^{2}.

$$9\sqrt{3}+5\sqrt{78}+2\sqrt{19}-2\sqrt{12}$$

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}.

$$9\sqrt{3}+5\sqrt{78}+2\sqrt{19}-2\times 2\sqrt{3}$$

Multiply -2 and 2 to get -4.

$$9\sqrt{3}+5\sqrt{78}+2\sqrt{19}-4\sqrt{3}$$

Combine 9\sqrt{3} and -4\sqrt{3} to get 5\sqrt{3}.

$$5\sqrt{3}+5\sqrt{78}+2\sqrt{19}$$