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Radicals Review − Guided Notes
Parts
[pic]
Simplest Radical Form
•
o The radical cannot be simplified further
• We always want our answers in __________________________________.
Getting Radicals into Simplest Radical Form
• Write the radicand as the product of factors, where ________________________ _________________________________________________________________.
• Take the square root of any perfect squares. _____________________________ _________________________________________________________________
• ___________________________________________________from the radicand.
Tips for Getting Radicals into Simplest Radical Form
• Always check if the radicand is a perfect square!
• Check if the radicand is factorable by common perfect squares − ______________
• _________________________________________________________, then it's in simplest radical form
•
o You don’t have to find the largest perfect square the first time you factor the radicand
Examples
Your Turn:
For problems 1 − 8, write the radical in simplest radical form.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
Adding and Subtracting Radicals
•
• Rewrite all radicals in __________________________________________!
• Add or subtract _____________________________ of like radical terms
Examples
Your Turn:
For problems 9 − 12, simplify. Write the answer in simplest radical form.
9. [pic] 10. [pic]
11. [pic] 12. [pic]
Multiplying Radicals
• Multiply like _________________
•
•
Write answer in simplest radical form
Examples
Your Turn:
For problems 13 − 16, simplify. Write the answer in simplest radical form.
13. [pic] 14. [pic]
15. [pic] 16. [pic]
Rationalizing Fractions
What is rationalizing?
•
• We generally rationalize _________________________. (But we can rationalize the numerator.)
Definitions:
• Monomial −
• Binomial −
• Conjugates −
• Examples:
o
o
•
(3 + x) and (3 – x) (4y5 + 2x2) and (4y5 – 2x2)
Your Turn:
For problems 17 – 20, identify the conjugate of the given expression. Then find the product of the conjugates.
17. [pic] 18. [pic]
19. [pic] 20. [pic]
Rationalizing the Denominator – Steps:
Is the denominator a monomial or a binomial?
Monomial
Step 1: Identify the denominator.
Step 2: Multiply the expression by 1.
Step 3: Rewrite 1 as
Denominator
Denominator
Step 4: Multiply the two fractions.
Step 5: Simplify the numerator and denominator, cancelling terms if necessary.
Binomial
Step 1: Identify the conjugate of the denominator.
Step 2: Multiply the expression by 1.
Step 3: Rewrite 1 as
Conjugate
Conjugate
Step 4: Multiply the two fractions. (Remember to FOIL!)
Step 5: Simplify the numerator and denominator, cancelling terms if necessary.
Example Monomial Problem:
|Algebra |Description of Steps |
| |Monomial |
| | |
|1. |Identify the denominator. |
| | |
| | |
| | |
|2. |Multiply the expression by 1. |
| | |
| | |
| | |
|3. |Rewrite 1 as |
| | |
| |Denominator |
| |Denominator |
| | |
|4. |Multiply the two fractions. |
| | |
| | |
| | |
| | |
| | |
| | |
|5. |Simplify the numerator and denominator, cancelling |
| |terms if necessary. |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
Practice Problems:
21. [pic] 22. [pic]
23. [pic] 24. [pic]
25. [pic] 26. [pic]
Example Binomial Problem:
|Algebra |Description of Steps |
| |Binomial |
| | |
|1. |Identify the conjugate of the denominator. |
| | |
| | |
| | |
|2. |Multiply the expression by 1. |
| | |
| | |
| | |
|3. |Rewrite 1 as |
| | |
| |Conjugate |
| |Conjugate |
| | |
|4. |Multiply the two fractions. |
| | |
| | |
| | |
| | |
| | |
| | |
|5. |Simplify the numerator and denominator, cancelling |
| |terms if necessary. |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
Practice Problems:
27. [pic] 28. [pic]
29. [pic] 30. [pic]
31. [pic]
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