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

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

Evaluate

$$\frac{16\sqrt{3}}{5}\approx 5.542562584$$

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}.

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

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{6}+4\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

Combine 12\sqrt{6} and 4\sqrt{6} to get 16\sqrt{6}.

$$\frac{16\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

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}.

$$\frac{16\sqrt{6}}{4\times 2\sqrt{2}-3\sqrt{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{16\sqrt{6}}{8\sqrt{2}-3\sqrt{2}}$$

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

$$\frac{16\sqrt{6}}{5\sqrt{2}}$$

Rationalize the denominator of \frac{16\sqrt{6}}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{16\sqrt{6}\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{16\sqrt{6}\sqrt{2}}{5\times 2}$$

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}.

$$\frac{16\sqrt{2}\sqrt{3}\sqrt{2}}{5\times 2}$$

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

$$\frac{16\times 2\sqrt{3}}{5\times 2}$$

Multiply 5 and 2 to get 10.

$$\frac{16\times 2\sqrt{3}}{10}$$

Multiply 16 and 2 to get 32.

$$\frac{32\sqrt{3}}{10}$$

Divide 32\sqrt{3} by 10 to get \frac{16}{5}\sqrt{3}.

$$\frac{16}{5}\sqrt{3}$$