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If you type 72\sqrt{72} into a calculator, you get 8.48528137...8.48528137... — an endless decimal that is impossible to write down completely. But in calculus and coordinate geometry, rounding destroys precision, so we write roots in exact radical form instead.
A surd is an exact root of a rational number that cannot be written as a clean fraction or integer, such as 2\sqrt{2} or 53\sqrt[3]{5}.
The word comes from the Latin surdus (meaning "deaf" or "mute"), translated in medieval Arabic and Latin texts to describe numbers whose exact values could not be expressed as a ratio of whole numbers.

Anatomy of a Radical

Before we pull factors out of a root, let's establish the four parts of any radical expression.
The radicand is the value inside the symbol, the index is the degree of the root (defaulting to 22 for square roots), and the coefficient is the multiplier out front.
How do we split a single giant radicand into smaller, simpler pieces?

The Product Rule of Radicals

The Product Rule of Radicals states that the root of a product equals the product of the individual roots: a⋅bn=an⋅bn\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} for non-negative real values.
This rule works because radicals are fractional powers: (ab)1/n=a1/n⋅b1/n(ab)^{1/n} = a^{1/n} \cdot b^{1/n}, so distributing the radical across a product follows directly from the laws of exponents.
An entire surd has all values under the radical (like 72\sqrt{72}), whereas a mixed surd has an extracted integer coefficient (like 626\sqrt{2}).