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Radicals Review − Guided Notes

Parts

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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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