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If you type 72\sqrt{72} into a calculator, you get a messy, non-repeating decimal like 8.48528137...8.48528137...
In algebra, physics, and higher math, we keep our answers precise by writing this in simplest radical form as 626\sqrt{2}.
The number sitting under the root symbol is called the radicand, and our goal is to extract every hidden square factor from it.
In 1525, German mathematician Christoff Rudolff introduced the modern \sqrt{} symbol in his algebra text Coss, styling it after a lowercase 'r' for radix, the Latin word for root.
How do we split a single radical into smaller pieces without altering its mathematical value?

The Product Property of Radicals

The product property of square roots states that for any non-negative numbers aa and bb, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
This property works because taking a square root is identical to raising a quantity to the power of 12\frac{1}{2}, and exponent rules dictate that (a⋅b)1/2=a1/2⋅b1/2(a \cdot b)^{1/2} = a^{1/2} \cdot b^{1/2}.
How can we use this multiplication rule to systematically break down a number like 72\sqrt{72}?

Step-by-Step Simplification

To simplify 72\sqrt{72}, find its greatest perfect square factor — a number from the sequence 4,9,16,25,36,49,64...4, 9, 16, 25, 36, 49, 64... that divides evenly into 7272.