Solve 1/6sqrt{35}+1/6-sqrt{35}

$$\frac{ 1 }{ 6 \sqrt{ 35 } } + \frac{ 1 }{ 6- \sqrt{ 35 } }$$

Evaluate

$$\frac{211\sqrt{35}}{210}+6\approx 11.944251592$$

Solution Steps

Rationalize the denominator of \frac{1}{6\sqrt{35}} by multiplying numerator and denominator by \sqrt{35}.

$$\frac{\sqrt{35}}{6\left(\sqrt{35}\right)^{2}}+\frac{1}{6-\sqrt{35}}$$

The square of \sqrt{35} is 35.

$$\frac{\sqrt{35}}{6\times 35}+\frac{1}{6-\sqrt{35}}$$

Multiply 6 and 35 to get 210.

$$\frac{\sqrt{35}}{210}+\frac{1}{6-\sqrt{35}}$$

Rationalize the denominator of \frac{1}{6-\sqrt{35}} by multiplying numerator and denominator by 6+\sqrt{35}.

$$\frac{\sqrt{35}}{210}+\frac{6+\sqrt{35}}{\left(6-\sqrt{35}\right)\left(6+\sqrt{35}\right)}$$

Consider \left(6-\sqrt{35}\right)\left(6+\sqrt{35}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{35}}{210}+\frac{6+\sqrt{35}}{6^{2}-\left(\sqrt{35}\right)^{2}}$$

Square 6. Square \sqrt{35}.

$$\frac{\sqrt{35}}{210}+\frac{6+\sqrt{35}}{36-35}$$

Subtract 35 from 36 to get 1.

$$\frac{\sqrt{35}}{210}+\frac{6+\sqrt{35}}{1}$$

Anything divided by one gives itself.

$$\frac{\sqrt{35}}{210}+6+\sqrt{35}$$

Combine \frac{\sqrt{35}}{210} and \sqrt{35} to get \frac{211}{210}\sqrt{35}.

$$\frac{211}{210}\sqrt{35}+6$$

Factor

$$\frac{211 \sqrt{35} + 1260}{210} = 11.944251591590568$$