Simplifying Square Roots


There are 2 parts to a square root.

  1. the RADICAL
  2. the RADICAND

The radical tells us that we’re looking for identical factors.

The radicand is a number that needs to be factored so we can pull out the identical ones.


Example: \sqrt{25}=\sqrt{5\cdot5} \Longrightarrow since there are 2 5’s \Longrightarrow \sqrt{5\cdot5}=5




Simplify \sqrt{72t^5}

Step 1: factor the radicand

72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

t^5 = t \cdot t \cdot t \cdot t \cdot t

Step 2: pull out factors that a repeated twice.

\mathbf{2} \cdot \mathbf{2} \cdot  2 \cdot \mathbf{3} \cdot \mathbf{3} <code><span class="pun">

\mathbf{t} \cdot \mathbf{t} \cdot \mathbf{t} \cdot \mathbf{t} \cdot t

Step 3: Doubled factors are removed from the radical and made single. All other factors stay inside the radical.

\mathbf{2} \cdot \mathbf{3} \cdot\mathbf{t} \cdot \mathbf{t} \sqrt{2t}

Step 4: Simplify