AI Tutor
If you simplify 72\sqrt{72} by factoring out 44, you get 2182\sqrt{18} — but you aren't done, because 1818 still has a square factor inside it.
Finding the largest perfect square factor right away lets you simplify any radical completely in a single step instead of doing multiple rounds of reduction.
What does our toolkit of perfect squares actually look like?

The Perfect Square Reference Strip

A perfect square is the product of an integer multiplied by itself, such as 6×6=366 \times 6 = 36.
How do we efficiently check which of these squares divides into our radicand without endless trial and error?

The Top-Down Countdown Method

To find the largest square factor of a number NN, divide NN by 22 and start testing the perfect squares below that midpoint, counting downward.
Let's apply this top-down search to find the greatest perfect square factor of 108108.
First, take half of 108108, which is 5454; the perfect squares below 5454 in descending order are 49,36,25,16,9,49, 36, 25, 16, 9, and 44.
Testing down the list: 108÷49108 \div 49 is not an integer, but 108÷36=3108 \div 36 = 3, meaning 3636 is the largest perfect square factor of 108108.