Solve frac{2sqrt{54}+8sqrt{6}}{6sqrt{12}-5sqrt{3}}

$$\frac { 2 \sqrt { 54 } + 8 \sqrt { 6 } } { 6 \sqrt { 12 } - 5 \sqrt { 3 } }$$

Evaluate

$$2\sqrt{2}\approx 2.828427125$$

Solution Steps

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{2\times 3\sqrt{6}+8\sqrt{6}}{6\sqrt{12}-5\sqrt{3}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{6}+8\sqrt{6}}{6\sqrt{12}-5\sqrt{3}}$$

Combine 6\sqrt{6} and 8\sqrt{6} to get 14\sqrt{6}.

$$\frac{14\sqrt{6}}{6\sqrt{12}-5\sqrt{3}}$$

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}.

$$\frac{14\sqrt{6}}{6\times 2\sqrt{3}-5\sqrt{3}}$$

Multiply 6 and 2 to get 12.

$$\frac{14\sqrt{6}}{12\sqrt{3}-5\sqrt{3}}$$

Combine 12\sqrt{3} and -5\sqrt{3} to get 7\sqrt{3}.

$$\frac{14\sqrt{6}}{7\sqrt{3}}$$

Cancel out 7 in both numerator and denominator.

$$\frac{2\sqrt{6}}{\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{6}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{6}\sqrt{3}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{2\sqrt{3}\sqrt{2}\sqrt{3}}{3}$$

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

$$\frac{2\times 3\sqrt{2}}{3}$$

Cancel out 3 and 3.

$$2\sqrt{2}$$