Solve frac{6sqrt{5}}{-3sqrt{8}+3sqrt{32}+4sqrt{50}}

$$\frac { 6 \sqrt { 5 } } { - 3 \sqrt { 8 } + 3 \sqrt { 32 } + 4 \sqrt { 50 } }$$

Evaluate

$$\frac{3\sqrt{10}}{26}\approx 0.364878192$$

Solution Steps

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}.

$$\frac{6\sqrt{5}}{-3\times 2\sqrt{2}+3\sqrt{32}+4\sqrt{50}}$$

Multiply -3 and 2 to get -6.

$$\frac{6\sqrt{5}}{-6\sqrt{2}+3\sqrt{32}+4\sqrt{50}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{6\sqrt{5}}{-6\sqrt{2}+3\times 4\sqrt{2}+4\sqrt{50}}$$

Multiply 3 and 4 to get 12.

$$\frac{6\sqrt{5}}{-6\sqrt{2}+12\sqrt{2}+4\sqrt{50}}$$

Combine -6\sqrt{2} and 12\sqrt{2} to get 6\sqrt{2}.

$$\frac{6\sqrt{5}}{6\sqrt{2}+4\sqrt{50}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{6\sqrt{5}}{6\sqrt{2}+4\times 5\sqrt{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{6\sqrt{5}}{6\sqrt{2}+20\sqrt{2}}$$

Combine 6\sqrt{2} and 20\sqrt{2} to get 26\sqrt{2}.

$$\frac{6\sqrt{5}}{26\sqrt{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\sqrt{5}}{13\sqrt{2}}$$

Rationalize the denominator of \frac{3\sqrt{5}}{13\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{5}\sqrt{2}}{13\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{5}\sqrt{2}}{13\times 2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{10}}{13\times 2}$$

Multiply 13 and 2 to get 26.

$$\frac{3\sqrt{10}}{26}$$