Solve 7/5-2sqrt{6}

$$\frac { 7 } { 5 - 2 \sqrt { 6 } }$$

Evaluate

$$14\sqrt{6}+35\approx 69.292856399$$

Solution Steps

Rationalize the denominator of \frac{7}{5-2\sqrt{6}} by multiplying numerator and denominator by 5+2\sqrt{6}.

$$\frac{7\left(5+2\sqrt{6}\right)}{\left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right)}$$

Consider \left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{7\left(5+2\sqrt{6}\right)}{5^{2}-\left(-2\sqrt{6}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{7\left(5+2\sqrt{6}\right)}{25-\left(-2\sqrt{6}\right)^{2}}$$

Expand \left(-2\sqrt{6}\right)^{2}.

$$\frac{7\left(5+2\sqrt{6}\right)}{25-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}$$

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

$$\frac{7\left(5+2\sqrt{6}\right)}{25-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{7\left(5+2\sqrt{6}\right)}{25-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{7\left(5+2\sqrt{6}\right)}{25-24}$$

Subtract 24 from 25 to get 1.

$$\frac{7\left(5+2\sqrt{6}\right)}{1}$$

Anything divided by one gives itself.

$$7\left(5+2\sqrt{6}\right)$$

Use the distributive property to multiply 7 by 5+2\sqrt{6}.

$$35+14\sqrt{6}$$