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If you calculate 50\sqrt{50}, you get an endless decimal like 7.071067…7.071067\dots, but in algebra, we keep numbers exact by writing 525\sqrt{2} instead. A perfect square is an integer produced by multiplying a whole number by itself, such as 44, 99, 1616, or 2525.
How do we split 50\sqrt{50} to reveal that hidden whole number?

The Product Property of Radicals

The Product Property of Radicals states that for non-negative numbers, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}. This works because square roots undo exponentiation: (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.
What happens if a number has more than one perfect square factor hidden inside it?

Extracting the Largest Factor

For 72\sqrt{72}, notice that both 44, 99, and 3636 divide evenly into 7272. Always choose the largest single perfect square factor (3636) so you simplify completely in one step: 72=36⋅2=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.
Be careful never to split radicals over addition: 25+9=34≈5.83\sqrt{25 + 9} = \sqrt{34} \approx 5.83, which is not equal to 25+9=8\sqrt{25} + \sqrt{9} = 8. To simplify any square root quickly, test perfect squares starting just below half your radicand.

Key takeaways

  • The Product Property of Square Roots states that for any non-negative real numbers aa and bb, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
  • A numerical square root expression is in simplest radical form when the radicand contains no factors (other than 11) that are perfect squares.