Solve -1/1-2sqrt{11}

$$\frac { - 1 } { 1 - 2 \sqrt { 11 } }$$

Evaluate

$$\frac{2\sqrt{11}+1}{43}\approx 0.177517432$$

Solution Steps

Rationalize the denominator of \frac{-1}{1-2\sqrt{11}} by multiplying numerator and denominator by 1+2\sqrt{11}.

$$\frac{-\left(1+2\sqrt{11}\right)}{\left(1-2\sqrt{11}\right)\left(1+2\sqrt{11}\right)}$$

Consider \left(1-2\sqrt{11}\right)\left(1+2\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-\left(1+2\sqrt{11}\right)}{1^{2}-\left(-2\sqrt{11}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{-\left(1+2\sqrt{11}\right)}{1-\left(-2\sqrt{11}\right)^{2}}$$

Expand \left(-2\sqrt{11}\right)^{2}.

$$\frac{-\left(1+2\sqrt{11}\right)}{1-\left(-2\right)^{2}\left(\sqrt{11}\right)^{2}}$$

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

$$\frac{-\left(1+2\sqrt{11}\right)}{1-4\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{-\left(1+2\sqrt{11}\right)}{1-4\times 11}$$

Multiply 4 and 11 to get 44.

$$\frac{-\left(1+2\sqrt{11}\right)}{1-44}$$

Subtract 44 from 1 to get -43.

$$\frac{-\left(1+2\sqrt{11}\right)}{-43}$$

To find the opposite of 1+2\sqrt{11}, find the opposite of each term.

$$\frac{-1-2\sqrt{11}}{-43}$$

Multiply both numerator and denominator by -1.

$$\frac{1+2\sqrt{11}}{43}$$