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What if you simplify 72\sqrt{72} down to 2182\sqrt{18}, but your teacher marks it incomplete?
Even though 2182\sqrt{18} equals 72\sqrt{72}, the radicand (the number sitting inside the radical symbol) still contains another hidden square: 99.
How do we ensure every single square factor is completely extracted from under the radical?

The Product Rule for Radicals

Since (u⋅v)2=u2⋅v2(u \cdot v)^2 = u^2 \cdot v^2, undoing a square distributes across multiplication. That gives us the product rule for radicals: a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} for any non-negative numbers aa and bb.
What happens when a radicand contains more than one square factor, such as both 44 and 99?

Method 1: Multi-Step Extraction

If you don't immediately see the largest square, you can pull out smaller square factors one at a time. Each root you extract becomes a coefficient (a multiplier out front) and multiplies whatever is already outside.
Let's simplify 720\sqrt{720}. First, we notice 44 divides into 720720, giving 4⋅180=2180\sqrt{4 \cdot 180} = 2\sqrt{180}.
Each step chipped away a perfect square (44, then 99, then another 44), multiplying the outside factors together: 2×3×2=122 \times 3 \times 2 = 12.
Can we reach the final answer in a single step instead of three?