AI Tutor
If you simplify 72\sqrt{72} to 2182\sqrt{18} and your friend simplifies it to 383\sqrt{8}, who is right?
Surprisingly, both answers are incomplete because the number 7272 hides more than one perfect square factor inside.
How do these multiple square factors work together, and how can we guarantee our radical is fully simplified?

The Product Rule and Why It Works

A radicand is simply the value sitting under the radical sign \sqrt{}.
The product rule for radicals states that a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} for any non-negative numbers aa and bb.
This rule works because squaring both sides gives (ab)2=(a)2(b)2=a⋅b(\sqrt{a}\sqrt{b})^2 = (\sqrt{a})^2(\sqrt{b})^2 = a \cdot b, proving that multiplying square roots directly tracks multiplying their radicands. In 1525, mathematician Christoff Rudolff introduced the \sqrt{} symbol in his book Coss, modeling it after a lowercase 'r' for radix, the Latin word for root.
What happens when a radicand contains several perfect square factors that you extract one step at a time?

Method 1: Step-by-Step Extraction

You do not need to find the single biggest perfect square immediately; you can pull out smaller square factors sequentially.
If you notice that 44 divides into 7272, rewrite 72\sqrt{72} as 4⋅18=218\sqrt{4 \cdot 18} = 2\sqrt{18}.
Since 1818 still contains the square factor 99, extract it too: 218=29⋅2=2⋅32=622\sqrt{18} = 2\sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}.