Download √16 ? 98 2 49 7 7 1 7 7 7 49 2 98

7.1 Prime, Composite, Simplify Radicals
Name _______________________________
Definitions:
Prime number___________________________________________________________
Composite Number _______________________________________________________
Prime Factorization _______________________________________________________
1. Determine the prime numbers.
To prime factor a number, use the tree method or the Birthday Cake method
Tree Method
Birthday Cake Method
1
98
2
7
49
7
7
7 49
7
2
98
2. Prime factor each of the following:
24 _____________
600 ____________
108_____________ 1000 ___________
3. What are Roots? What is the difference between 2 16
and
√16 ?
2
4. “Roots” or radicals are the inverse operation of applying exponents. You can undo a power with a
radical and a radical can undo a power. For example, 32 = 9, 𝑠𝑜 √9 = 3 and 43 = 64, 𝑠𝑜 √64 = 4
3
If the index is not written, then the
degree is 2. √25 = 5
√49 = 7
5. How do you find roots?
A. Nice roots can be found by memory or by using a calculator.
√25
√81
3
√125
4
√81
4
√4096
B. Not nice roots can be found using your calculator and rounding which gives you an
approximate answer or can be simplified which gives you an exact answer.
Exact Value vs Approximate Value
6. You can determine the approximate value of √75 . Determine the perfect square that is
closest to but less than 75. Then determine the perfect square that is closest to but greater
than 75.
√16 √25 √36 √49 √64 √81 √100 √121 √144
√64 = 8
√75 = ?
√81 = 9
Now you know that √75 is between 8 and 9, and you can use your calculator to calculate the
approximate square root as 8.660254038 …
There are times when an exact solution is necessary or you need to simplify a calculation and a
decimal answer makes it difficult to simplify, so we simplify the radical instead of calculating
the decimal value.
√75
√5 ∙ 5 ∙ 3 Rewrite 75 in prime factored form
√5 ∙ 5 ∙ 3 since the index is 2 because you are finding the square root, circle any groups of 2
numbers.
5√3 is the exact simplified form of √75 because one of the numbers in the circled group is
written outside the radical and the other crossed off. Any uncircled number is left under the
radical symbol.
7. BEFORE YOU SIMPLIFY, CHECK WITH YOUR CALCULATOR TO SEE IF THE RADICAL IS
A NICE SQUARE ROOT OR THIRD ROOT OR NTH ROOT. If you get decimals, then simplify
using the following steps.
Steps for Simplifying Radicals
1. Write the prime factorization of your radicand and determine the index of the radical.
2. If the index is 2, circle groups of 2 identical numbers or variables. If the index is 3,
circle groups of 3 identical numbers or variables.
3. The number or variable from each circled group will show up outside the radical symbol 1
time and the rest of the circled group will be crossed off.
4. Anything left uncircled will remain under the radical. If everything under the radical
symbol is circled, the radical symbol will disappear.
5. Finish simplifying by multiplying the numbers and variables outside the radical together.
Do together as a class
√600
Do Yourself
√72
√120
√32
√125𝑥 3 𝑦 2
√36𝑥 2 𝑦 3
5√168𝑥 7
8𝑦√75𝑦 5
3
4
√56𝑥 5
√144𝑦 6
8. What if there is a number in front of the radical symbol? 5√32