Solve 112/12+sqrt{7}-9

$$\frac{ 112 }{ 12+ \sqrt{ 7 } } -9$$

Evaluate

$$\frac{111-112\sqrt{7}}{137}\approx -1.352730999$$

Short Solution Steps

Rationalize the denominator of \frac{112}{12+\sqrt{7}} by multiplying numerator and denominator by 12-\sqrt{7}.

$$\frac{112\left(12-\sqrt{7}\right)}{\left(12+\sqrt{7}\right)\left(12-\sqrt{7}\right)}-9$$

Consider \left(12+\sqrt{7}\right)\left(12-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{112\left(12-\sqrt{7}\right)}{12^{2}-\left(\sqrt{7}\right)^{2}}-9$$

Square 12. Square \sqrt{7}.

$$\frac{112\left(12-\sqrt{7}\right)}{144-7}-9$$

Subtract 7 from 144 to get 137.

$$\frac{112\left(12-\sqrt{7}\right)}{137}-9$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9 times \frac{137}{137}.

$$\frac{112\left(12-\sqrt{7}\right)}{137}-\frac{9\times 137}{137}$$

Since \frac{112\left(12-\sqrt{7}\right)}{137} and \frac{9\times 137}{137} have the same denominator, subtract them by subtracting their numerators.

$$\frac{112\left(12-\sqrt{7}\right)-9\times 137}{137}$$

Do the multiplications in 112\left(12-\sqrt{7}\right)-9\times 137.

$$\frac{1344-112\sqrt{7}-1233}{137}$$

Do the calculations in 1344-112\sqrt{7}-1233.

$$\frac{111-112\sqrt{7}}{137}$$