Solve 4/3sqrt{32-sqrt{50}}

$$\frac { 4 } { 3 \sqrt { 32 } - \sqrt { 50 } }$$

Evaluate

$$\frac{2\sqrt{2}}{7}\approx 0.404061018$$

Solution Steps

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}.

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

Multiply 3 and 4 to get 12.

$$\frac{4}{12\sqrt{2}-\sqrt{50}}$$

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}.

$$\frac{4}{12\sqrt{2}-5\sqrt{2}}$$

Combine 12\sqrt{2} and -5\sqrt{2} to get 7\sqrt{2}.

$$\frac{4}{7\sqrt{2}}$$

Rationalize the denominator of \frac{4}{7\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{4\sqrt{2}}{7\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{2}}{7\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\sqrt{2}}{7}$$