Solve frac{1/3+sqrt{318}}{sqrt{32}}

$$\frac { \frac { 1 } { 3 } + \sqrt { 318 } } { \sqrt { 32 } }$$

Evaluate

$$\frac{\sqrt{2}+6\sqrt{159}}{24}\approx 3.211305618$$

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}.

$$\frac{\frac{1}{3}+\sqrt{318}}{4\sqrt{2}}$$

Rationalize the denominator of \frac{\frac{1}{3}+\sqrt{318}}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(\frac{1}{3}+\sqrt{318}\right)\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\frac{1}{3}+\sqrt{318}\right)\sqrt{2}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(\frac{1}{3}+\sqrt{318}\right)\sqrt{2}}{8}$$

Use the distributive property to multiply \frac{1}{3}+\sqrt{318} by \sqrt{2}.

$$\frac{\frac{1}{3}\sqrt{2}+\sqrt{318}\sqrt{2}}{8}$$

Factor 318=2\times 159. Rewrite the square root of the product \sqrt{2\times 159} as the product of square roots \sqrt{2}\sqrt{159}.

$$\frac{\frac{1}{3}\sqrt{2}+\sqrt{2}\sqrt{159}\sqrt{2}}{8}$$

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

$$\frac{\frac{1}{3}\sqrt{2}+2\sqrt{159}}{8}$$

Factor

$$\frac{\sqrt{2} + 6 \sqrt{159}}{24} = 3.211305618328502$$