**Middle School Number Sense Lesson 69: Square Root Simplification**

About 5 months ago, we discussed

**estimating**square roots (

**EstSquareRoot**). About 4 months ago, we

**calculated**them (

**SquareRoot**). Today we will

**simplify**square roots. This concept appeared

**5 times**last year on middle school tests, with a median placement at

**question # 67**.

**Please Like our Facebook Page (**

**https://www.facebook.com/numberdojo/**

**) if you want to see new posts on your wall. I will reward you with a free concept index or flashcard file of your choice. Thank you!**

**Number Dojo Level: 244**

These questions require a fair amount of mental manipulation. For each one, there will be a square root that will not simplify completely (in other words, it will not be a perfect square). But at least one factor of the number under the

**radicand**

**(square root sign)**

**will be**a perfect square. Our task is to factor out that perfect square and end up with the answer in its simplest form.

**How to Simplify Square Roots:**

- Factor the number under the radicand until you have a perfect square multiplied by some other number.
- Take the square root of that perfect square as you pull it out of (and place it in front of) the radicand. Leave the other factor inside.
- Repeat Steps 1 and 2 if possible.

**Example 1: If √45 simplifies as a√b, then a = ___**

- Factor 45 to
**9 x 5**. - Take the square root of
**9**to get**3**. Place the 3 in front of the radicand and the 5 inside. You have**3√5**. - 5 does not have any factors which are perfect squares.
- The question asked for
**a**, which is**3**.

**Example 2: If √98 simplifies as a√b, then a = ___**

- 98 can be factored to
**49 x 2**. **√49**is**7**. Place this in front of the radicand and the 2 inside. You have**7****√2**.- 2 doesn't have any other factors which are perfect squares.
- The question asked for
**a**, which is**7**.

**Example 3: If √(4 x 8 x 12) = r√s, then s = ___**

**(4 x 8 x 12)**is the same as**(32 x 2 x 6)**. This is**(64 x 6)**.**√64 = 8**. Put this in front to make**8√6**.**√6**doesn't simplify any further. The answer is**6**.

**Example 4: √12 + √75 = √x. Find x.**

- √12 simplifies to
**2√3**. - √75 simplifies to
**5√3**. - 2√3 + 5√3 =
**7√3**. - Since the question asked for everything to be under the radicand, square 7 to get
**49**. Multiply this by 3 to get**147**. The answer is**147**.

**Example 5: If √(5 x 10 x 25) simplifies to r√s, then r + s = ___**

- √(5 x 10 x 25) = √(50 x 25) = √(2 x 25 x 25). This simplifies to
**25√2**. - The question asked for
**r + s**. 25 + 2 =**27**.

**Here is a free worksheet to help you practice SquareRootSim:**

squarerootsim.pdf |

**Up Next for Middle School: Mult101var**