Part 4: Simplifying Square Roots_L08_U02_C3 Math_Ready for Review

Try it out! 
Simplify each expression by factoring out the perfect squares.

$$\begin{gathered} 1) \sqrt{45} \\ 2) \sqrt{72} \\ 3) \sqrt{98} \\ 4) \sqrt{243} \end{gathered}$$

***Solution text below.

$1) \sqrt{45}$Factors of 45: 1, 3, 5, 9, 15, 45
9 is the only factor of 45 that is a perfect square.
$$\begin{gathered} \sqrt{45} \\ \sqrt{9}\times \sqrt{5} \\ 3\sqrt{5} \end{gathered}$$
$2) \sqrt{72}$Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
4,9, and 36 are all perfect squares.
When more than one factor is a perfect square, factor out the largest perfect square.
36 is the largest perfect square.
$$\begin{gathered} \sqrt{72} \\ \sqrt{36}\times \sqrt{2} \\ 6\sqrt{2} \end{gathered}$$
$3) \sqrt{98}$Factors of 98: 1, 2, 7, 14, 49, 98
49 is the only factor that is a perfect square.
$$\begin{gathered} \sqrt{49}\times \sqrt{2} \\ 7\sqrt{2} \end{gathered}$$
$4) \sqrt{243}$Factors of 243: 1, 3, 9, 27, 81, 243
9 and 81 are perfect squares.
When more than one factor is a perfect square, factor out the largest perfect square.
81 is the largest perfect square.
$$\begin{gathered} \sqrt{243} \\ \sqrt{81}\times \sqrt{3} \\ 9\sqrt{3} \end{gathered}$$
***End solution text

Prime Factorization

Prime factorization is another way to simplify square roots.
This method is particularly helpful with imperfect squares that are not the product of perfect squares.

View the Simplifying Square Roots with Prime Factorization Video to learn more about this strategy.
While watching the video, complete the corresponding portion of your Lesson 8 Interactive Notes. 

**Insert video – Include an example that can not be simplified

Try it out! 
Step 1: Construct a factor tree to determine the prime factorization of each value below.
Step 2: Simplify each imperfect square using the prime factorization of each value.

$$\begin{gathered} 1) \sqrt{12} \\ 2) \sqrt{75} \\ 3) \sqrt{58} \\ 4) \sqrt{112} \end{gathered}$$

**Solution text below.

Compare your solutions to those shown below.

$1) \sqrt{12}$

Prime Factorization of 12: $2\times 2\times 3$
$$\begin{gathered} \sqrt{2\times 2\times 3} \\ \sqrt{\overline{)2\times 2}\times 3} \\ \sqrt{2^2\times 3} \\ 2\sqrt{3} \end{gathered}$$
$2) \sqrt{75}$

Prime Factorization of 75: $3\times 5\times 5$
$$\begin{gathered} \sqrt{3\times 5\times 5} \\ \sqrt{3\times \overline{)5\times 5}} \\ \sqrt{3\times 5^2} \\ 5\sqrt{3} \end{gathered}$$
$3) \sqrt{58}$Prime Factorization of 58: $2\times 29$There are no like factors to group.
The square root of 58 can not be simplified.

$4) \sqrt{112}$Prime Factorization of 112: $2\times 2\times 2\times 2\times 7$

$$\begin{gathered} \sqrt{2\times 2\times 2\times 2\times 7} \\ \sqrt{\overline{)2\times 2}\times \overline{)2\times 2}\times 7} \\ \sqrt{2^2\times 2^2\times 7} \\ 2·2\sqrt{7} \\ 4\sqrt{7} \end{gathered}$$
**end solution text

**Instructor Notes – Text will appear when button is clicked

How to Support Students:

The following activities can be used to support students that need additional practice or enrichment on estimating roots.

• Matching
  
  • Students will match an imperfect square to its simplified form.