Simplifying Square Roots

There are 2 parts to a square root.

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 $\Longrightarrow \sqrt{5\cdot5}=5$

Procedure

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

$6t^2\sqrt{2t}$