Rationalize the denominator of 7/(root 5 - 2) and simplify the answer.

Step-by-step solution

1. 1

   Identify the conjugate. The denominator is a difference of a radical and an integer; its conjugate flips only the middle sign:

   $\text{conjugate of }\sqrt5-2\ \text{is}\ \sqrt5+2.$

2. 2

   Multiply top and bottom by the conjugate.

   $\frac{7}{\sqrt5-2}\cdot\frac{\sqrt5+2}{\sqrt5+2}=\frac{7\left(\sqrt5+2\right)}{\left(\sqrt5-2\right)\left(\sqrt5+2\right)}.$

3. 3

   Evaluate the denominator with the difference of squares.

   $\left(\sqrt5\right)^{2}-2^{2}=5-4=1.$

   This is the pleasant surprise in this problem: the denominator collapses to $1$, so the whole fraction vanishes.

4. 4

   Write the result without a denominator.

   $\frac{7\left(\sqrt5+2\right)}{1}=7\sqrt5+14.$

   Leaving the answer as $\dfrac{7\sqrt5+14}{1}$ is not wrong but is not simplified; drop the $1$.

5. 5

   Check numerically. $\sqrt5=2.2360680$, so the original is $7/0.2360680=29.6524758$, and $7(2.2360680)+14=15.6524758+14=29.6524758$. Identical to every digit shown.