1Cademy - Simplifying $$3 + \sqrt{32}$$

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How to Simplify a Square Root Using the Product Property

Example

Simplifying $3 + \sqrt{32}$

Simplify the expression $3 + \sqrt{32}$ , which is the sum of an integer and a square root.

Step 1 — Rewrite the radicand using its largest perfect square factor. The largest perfect square dividing $32$ is $16$ (since $4^2 = 16$ ):

$3 + \sqrt{32} = 3 + \sqrt{16 \cdot 2}$

Step 2 — Apply the Product Property. Split the radical:

$3 + \sqrt{16 \cdot 2} = 3 + \sqrt{16} \cdot \sqrt{2}$

Step 3 — Simplify the radical. Since $\sqrt{16} = 4$ :

$3 + \sqrt{16} \cdot \sqrt{2} = 3 + 4\sqrt{2}$

The expression $3 + 4\sqrt{2}$ cannot be simplified further because the integer $3$ and the radical term $4\sqrt{2}$ are not like terms. Attempting to add an integer to a radical is analogous to trying to add an integer to a variable — the two terms have fundamentally different forms and cannot be combined into a single term.

Updated 2026-05-01

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