Here the idea is that the formula comes from a right triangle in the first place, so you might as well make use of that fact. All of my examples will come from one of the two integer-sided right triangles:

The value of v/c will always be chosen to be the ratio of one of the legs of a triangle to its longest side. The square-root in the formula will then be:

Square root = (other leg/longest side)

 A space ship leaves the earth and accelerates to 4/5 the speed of light. The ship and the earth exchange regular radio messages so that the earth-bound observer can figure out (by correcting for message-travel time) the earth-times of events on the spaceship.

Two events on the ship are separated by three years of ship-time. How many earth-years separate the events?

The problem statement tells you that v/c = 4/5. Look at the 3-4-5 triangle. The leg not used yet is 3, so the square root in the formula must be equal to 3/5. The problem statement tells you that T = 3 years elapse on the ship. The earth-time is then 3 years divided by 3/5. That is the same as 3 years multiplied by 5/3 or 5 years.