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If you build a rectangle with sides measuring 2\sqrt{2} and 8\sqrt{8}, its area turns out to be a crisp, whole number: exactly 44.
To see why two messy square roots multiply into a clean integer, let's look at the parts of a radical expression: the radicand is the number under the root, and the coefficient is the number multiplying the root from the outside.
How do we combine radicals when multiplying them together?

The Product Property of Radicals

The Product Property of Radicals states that for any non-negative real numbers aa and bb, a⋅b=a⋅b\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}.
This rule works because a square root represents an exponent of 12\frac{1}{2}, and exponents distribute across multiplication: (a⋅b)1/2=a1/2⋅b1/2(a \cdot b)^{1/2} = a^{1/2} \cdot b^{1/2}.
Let's multiply 6⋅15\sqrt{6} \cdot \sqrt{15}. First, combine the radicands to get 90\sqrt{90}, then factor out the largest perfect square: 90=9⋅10=310\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}.
What happens when our radical terms have numbers sitting in front of them?

Multiplying with Coefficients

When coefficients are present, multiply the outside numbers with outside numbers and the inside numbers with inside numbers: (ab)⋅(cd)=(a⋅c)b⋅d(a\sqrt{b}) \cdot (c\sqrt{d}) = (a \cdot c)\sqrt{b \cdot d}.