Simplifying Radicals

Radicals (also called roots) are the opposite of exponents. This is also called an inverse.

    \[\sqrt[n]{a^n}=a\]

There are actually 3 parts to a radical. Square-roots don’t have an index because we assume it’s 2.

  1. the RADICAL
  2. the RADICAND
  3. the INDEX

The index tells us how many identical factors we’re looking for.

We know the Radical & Radicand from simplifying square roots.

 

Example: \sqrt[3]{125}=\sqrt{5\cdot5\cdot 5} \Longrightarrow since the index is 3, and there are 3- 5’s \Longrightarrow \sqrt[3]{5\cdot5\cdot5}=5

 

 

Procedure

Example: \sqrt[3]{72t^5}

To simplify radical expressions follow these steps.

 

Step 1: determine the index

The index is the small number in the angle of the radical symbol.

The index is 3.

Step 2: 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 3: pull out repeated factors of the amount of the index (in this case 3 times).

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

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

Step 4: repeated factors are removed from the radical and made single. All other factors stay inside the radical.

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

Step 5: Simplify

2t\sqrt[3]{9t^2}