Solve frac{2sqrt{27}-4sqrt{3}+sqrt{12}}{sqrt{75}+sqrt{48}-8sqrt{3}}

$$\frac { 2 \sqrt { 27 } - 4 \sqrt { 3 } + \sqrt { 12 } } { \sqrt { 75 } + \sqrt { 48 } - 8 \sqrt { 3 } }$$

Evaluate

$$4$$

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}.

$$\frac{2\times 3\sqrt{3}-4\sqrt{3}+\sqrt{12}}{\sqrt{75}+\sqrt{48}-8\sqrt{3}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{3}-4\sqrt{3}+\sqrt{12}}{\sqrt{75}+\sqrt{48}-8\sqrt{3}}$$

Combine 6\sqrt{3} and -4\sqrt{3} to get 2\sqrt{3}.

$$\frac{2\sqrt{3}+\sqrt{12}}{\sqrt{75}+\sqrt{48}-8\sqrt{3}}$$

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}.

$$\frac{2\sqrt{3}+2\sqrt{3}}{\sqrt{75}+\sqrt{48}-8\sqrt{3}}$$

Combine 2\sqrt{3} and 2\sqrt{3} to get 4\sqrt{3}.

$$\frac{4\sqrt{3}}{\sqrt{75}+\sqrt{48}-8\sqrt{3}}$$

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}.

$$\frac{4\sqrt{3}}{5\sqrt{3}+\sqrt{48}-8\sqrt{3}}$$

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}.

$$\frac{4\sqrt{3}}{5\sqrt{3}+4\sqrt{3}-8\sqrt{3}}$$

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

$$\frac{4\sqrt{3}}{9\sqrt{3}-8\sqrt{3}}$$

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

$$\frac{4\sqrt{3}}{\sqrt{3}}$$

Rationalize the denominator of \frac{4\sqrt{3}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{4\sqrt{3}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4\sqrt{3}\sqrt{3}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{4\times 3}{3}$$

Multiply 4 and 3 to get 12.

$$\frac{12}{3}$$

Divide 12 by 3 to get 4.

$$4$$

Factor

$$2^{2}$$