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If you build a square garden with an area of 72 sq ft72\text{ sq ft}, each side has a length of 72 ft\sqrt{72}\text{ ft}. A calculator spits out an endless decimal, 8.485281...8.485281..., but mathematicians prefer an exact, clean form: 626\sqrt{2}.
How do we pull a whole number out of a radical without changing its value? We search for perfect square factors hiding inside the number under the root.
The number underneath the radical sign is called the radicand. 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}.
Why does this rule work? Because squaring both sides gives (a⋅b)2=(a)2⋅(b)2=a⋅b(\sqrt{a} \cdot \sqrt{b})^2 = (\sqrt{a})^2 \cdot (\sqrt{b})^2 = a \cdot b, which matches (ab)2(\sqrt{ab})^2.
In 1525, German mathematician Christoff Rudolff introduced the modern radical symbol \sqrt{} in his algebra textbook Coss, likely styling it after a lowercase letter r for the Latin word radix, meaning root.
Finding factors is easy when the numbers are small, but how do we ensure we always catch the largest square without guessing?

The Largest Factor Strategy

A perfect square is the product of an integer multiplied by itself, such as 4,9,16,25,36,49,64,81,4, 9, 16, 25, 36, 49, 64, 81, and 100100.
To simplify n\sqrt{n}, start by finding half of the radicand, then test the list of perfect squares downward until you find one that divides nn evenly.
Let's test this method on 72\sqrt{72}. Since half of 7272 is 3636, and 3636 is itself a perfect square, 72÷36=272 \div 36 = 2, giving 36⋅2=62\sqrt{36 \cdot 2} = 6\sqrt{2} instantly.
A common mistake is stopping at the first small square you spot. If you chose 44, you would write 4⋅18=218\sqrt{4 \cdot 18} = 2\sqrt{18}, which is incomplete because 1818 still contains the square factor 99.