Solve frac{2sqrt{54}+4sqrt{6}}{4sqrt{8}-3sqrt{2}}

$$\frac { 2 \sqrt { 54 } + 4 \sqrt { 6 } } { 4 \sqrt { 8 } - 3 \sqrt { 2 } }$$

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

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}.

$$\frac{2\times 3\sqrt{6}+4\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{6}+4\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

Combine 6\sqrt{6} and 4\sqrt{6} to get 10\sqrt{6}.

$$\frac{10\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{10\sqrt{6}}{4\times 2\sqrt{2}-3\sqrt{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{10\sqrt{6}}{8\sqrt{2}-3\sqrt{2}}$$

Combine 8\sqrt{2} and -3\sqrt{2} to get 5\sqrt{2}.

$$\frac{10\sqrt{6}}{5\sqrt{2}}$$

Cancel out 5 in both numerator and denominator.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Cancel out 2 and 2.

$$2\sqrt{3}$$