Download Guided Notes Module 25 and more Exams Elementary Mathematics in PDF only on Docsity!
15.2 Simplifying Radicals:
Chapter 15 Section 2 Textbook pp 1025-1028.
1. Multiplication Property of Radicals:
Multiplication Property of Radicals
Let ๐๐ and ๐๐ represent real numbers such that โ
๐๐
๐๐
are both real. Then
๐๐
๐๐
๐๐
For example:
Simplified Form of a Radical:
Consider any radical expression where the radicand is written as a product of prime
factors. The expression is in simplified form if all the following conditions are met:
- The radicand has no factor raised to a power greater than or equal to the
index.
- The radicand does not contain a fraction.
- There are no radicals in the denominator of a fraction.
For example, the following radicals are not simplified.
- The expressions โ๐ฅ๐ฅ
2
fails rule 1.
- The expression
1
4
fails rule 2.
- The expression
1
โ
fails rule 3.
2. Simplifying Radicals by Using the Multiplication Property of Radicals:
The expression โ๐ฅ๐ฅ
2
is not simplified because it fails condition 1. Because ๐ฅ๐ฅ
2
is a perfect
square, โ๐ฅ๐ฅ
2
is easily simplified:
2
Guided Notes Module 25
Example 1 โ Using the Multiplication Property to Simplify a Radical Expression:
Simplify the expression assuming that ๐ฅ๐ฅ โฅ 0. โ๐ฅ๐ฅ
9
When we square root a number or variable, we are looking for sets of 2 to make a
root. For example: โ
2 โ 2 = 2 or โ
2 โ 2 โ 2 , because we only
have 1 set of 2โs, we can only pull one 2 out as a root, the other 2 has to stay
behind with the radical: โ8 = 2โ 2
So for the above example, we do not have sets of 2, we have 9 xโs; therefore, we are
going to have one left under the radical.
9
8
4
We divide 8 รท 2 = 4
Try โ #1:
Simplify the expression assuming that ๐ฅ๐ฅ โฅ 0. โ๐ฅ๐ฅ
11
Example 2 โ Using the Multiplication Property to Simplify a Radical Expression:
Convert each expression. Assume all variables represent positive real numbers.
a) โ๐๐
15
Do the same thing we did before, divide: 15 รท 2 = 7 , ๐ค๐ค๐ค๐ค๐ค๐คโ 1 ๐๐๐๐๐๐๐ค๐ค ๐๐๐๐๐๐๐๐:
7
b) ๏ฟฝ๐ฅ๐ฅ
2
5
Divide with both exponents: 2 รท 2 = 1, ๐ค๐ค๐ค๐ค๐ค๐คโ ๐๐๐๐ ๐๐๐๐๐๐๐๐๐ค๐ค๐๐๐๐๐๐๐๐
5 รท 2 = 4 ๐ค๐ค๐ค๐ค๐ค๐คโ ๐๐ ๐๐๐๐๐๐๐๐๐ค๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐ 1
2
Try โ #2:
Convert each expression. Assume all variables represent positive real numbers.
a) ๏ฟฝ๐ฆ๐ฆ
11
b) ๏ฟฝ๐ฅ๐ฅ
8
13
Try โ #5:
Simplify the expression. Assume the variables represent positive real numbers.
a) โ
b) โ 60 ๐ฅ๐ฅ
2
c) 7 โ 18 ๐ค๐ค
10
3. Simplifying Radicals by Using the Order Of Operations:
Example 6โ Using the Order of Operations to Simplify Radicals:
Use the order of operations to simplify the expressions. Assume ๐๐ > 0
a)
๐๐
7
๐๐
3
4
2
b) ๏ฟฝ
6
96
1
16
1
4
c)
27 ๐ฅ๐ฅ
5
3๐ฅ๐ฅ
4
2
Try โ #6:
Use the order of operations to simplify the expressions. Assume ๐๐ > 0
a)
๐ฃ๐ฃ
21
๐ฃ๐ฃ
5
b) ๏ฟฝ
8
50
c)
32๐ง๐ง
3
2๐ง๐ง
Example 7โ Using the Order of Operations:
Use the order of operations to simplify:
7 โ 50
15
Try:
Use the order of operations to simplify:
2 โ 300
30
For more examples please refer to page 1031-1033 and complete the following
problems: 15, 16, 17, 31, 32, 33, 39, 40, 41, 51, 52, 53, 59, 60, 61.
End of Guided Notes Module 25
Complete Module 25 ALEKS
