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Imagine facing 500\sqrt{500} on a geometry exam with no calculator in sight. How do you break down three- and four-digit numbers trapped under a radical sign without endlessly guessing factors?
The number inside the square root symbol is called the radicand, and simplifying it means pulling out any factor that is a perfect square.
How do we systematically pull out the largest possible square from larger numbers like 288 without doing unnecessary extra steps?

The Largest Perfect Square Strategy

The product property of radicals states that a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} because fractional exponents distribute across multiplication: (ab)1/2=a1/2b1/2(ab)^{1/2} = a^{1/2} b^{1/2}.
To simplify 288\sqrt{288}, test dividing by high perfect squares (144,121,100,81,64,49,36,25,16,9,4144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4) starting from half of 288 downward.
A common mistake is stopping early at 686\sqrt{8}. A radical expression is in simplest radical form only when the radicand contains no square factors other than 1.
What happens when radicands are well into the thousands, like 1,080\sqrt{1,080}, and testing individual squares becomes too slow?

The Prime Factorization Method

When numbers are too large to easily spot square factors, break the radicand down into its prime factorization and group identical prime factors in pairs.
Because p⋅p=p2=p\sqrt{p \cdot p} = \sqrt{p^2} = p, every matched pair of prime factors inside the radical translates to a single whole factor outside.