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If you punch 72\sqrt{72} into a calculator, you get 8.48528137...8.48528137... — a rounded decimal that loses its exact mathematical value. Writing it in simplified radical form lets you keep complete precision while making the number much easier to calculate with.
Before breaking numbers apart, let's identify the parts: the radicand is the value resting inside the root symbol, while a number multiplying the root from the outside is called a coefficient.
How do we pull whole numbers out of the radicand without altering the expression's actual value?

The Product Property of Radicals

The product property of radicals states that for any non-negative numbers aa and bb, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
This works because square roots are fractional exponents (x1/2x^{1/2}); since (ab)1/2=a1/2b1/2(ab)^{1/2} = a^{1/2} b^{1/2}, splitting the radicand into factors preserves equality.
What strategy guarantees that you simplify a root completely in a single step?

Factoring Out the Largest Perfect Square

To simplify 72\sqrt{72}, test perfect squares (4,9,16,25,36,…4, 9, 16, 25, 36, \dots) in descending order to find the greatest perfect square factor.
While 7272 is divisible by 44 and 99, picking 3636 gives 36⋅2=62\sqrt{36 \cdot 2} = 6\sqrt{2} immediately, leaving behind a radicand with no remaining square factors.