Solve 1/13-sqrt{168}

$$\frac { 1 } { 13 - \sqrt { 168 } }$$

Evaluate

$$2\sqrt{42}+13\approx 25.961481397$$

Solution Steps

Factor 168=2^{2}\times 42. Rewrite the square root of the product \sqrt{2^{2}\times 42} as the product of square roots \sqrt{2^{2}}\sqrt{42}. Take the square root of 2^{2}.

$$\frac{1}{13-2\sqrt{42}}$$

Rationalize the denominator of \frac{1}{13-2\sqrt{42}} by multiplying numerator and denominator by 13+2\sqrt{42}.

$$\frac{13+2\sqrt{42}}{\left(13-2\sqrt{42}\right)\left(13+2\sqrt{42}\right)}$$

Consider \left(13-2\sqrt{42}\right)\left(13+2\sqrt{42}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{13+2\sqrt{42}}{13^{2}-\left(-2\sqrt{42}\right)^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{13+2\sqrt{42}}{169-\left(-2\sqrt{42}\right)^{2}}$$

Expand \left(-2\sqrt{42}\right)^{2}.

$$\frac{13+2\sqrt{42}}{169-\left(-2\right)^{2}\left(\sqrt{42}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{13+2\sqrt{42}}{169-4\left(\sqrt{42}\right)^{2}}$$

The square of \sqrt{42} is 42.

$$\frac{13+2\sqrt{42}}{169-4\times 42}$$

Multiply 4 and 42 to get 168.

$$\frac{13+2\sqrt{42}}{169-168}$$

Subtract 168 from 169 to get 1.

$$\frac{13+2\sqrt{42}}{1}$$

Anything divided by one gives itself.

$$13+2\sqrt{42}$$