# chapter_6__cheat_sheet by ashrafp

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```									                                      Chapter 6 Cheat Sheet

Vocabulary:

Root:                                 Principal Root:

Big Idea:         For every _______________ there is a _________________

Ex:        52 = 25, _________ is a sq. root
53 = 125, _________ is a cube root
55 = 3125, _________ is a fifth root

If an = b, then a is an nth root of b

How many different roots?

If n is:          Odd: one
n
Even: and b is positive: 2 real roots (principal:       b and negative: - n b )
or if b is negative:   NO real roots

Note: If the problems says find ALL real roots, list principal AND negative
                 
If problems says find 4 x 2 , assume ONLY the principal (2x)

Examples:


What are the roots?

n
For any real number a,         an =      a if n is __________
Or |a| if n is ___________

Examples:       

More Problems: Pg. 364 1 – 6, 10-28 even, 33, 34, 42

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Chapter 6 Cheat Sheet

A radical is in simplest form if all perfect nth factors are reduced

1.   Does problem ask for all roots or only principal roots?
2.   Find “a” – the root so that n a n (use prime factors, or guess and check)
3.   Check the index
a. If index is ODD, n a n = a
b. If index is EVEN, n a n = |a|

4.   Remove unnecessary absolute value signs:
a. If root exponent is EVEN, drop the absolute value symbol

b. If root exponent is ODD, keep the absolute value


Examples:

n           n                              n         n         n
If       a and       b are real numbers, then       a *       b =       ab

Note: because these are REAL numbers, a and b MUST be non-negative.
    
Add to question 4 above:              
c. If exponent in either radicand is ODD, drop the absolute value symbol

(look at each variable separately):
Index is even
Root exponent is odd
Exponent in either original radicand is even

Examples:

More Problems: Pg. 371 10, 18, 19, 21, 29 – 31
6-2 Review W/S Multiplying Radical Expressions

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Chapter 6 Cheat Sheet

n
n           n                                  a         a
If       a and       b are real numbers, then   n
=   n
b         b

    
Examples:
     

Rationalizing the Denominator:
Goal – no radicals in the denominator, and no denominator in any radical.

Multiply numerator and denominator by a radical YOU CHOOSE. Make each
component numbers that are perfect to the nth power.

Examples:

More Problems: Pg. 371 37 – 42
6 -2 Review W/S, Multiplying and Dividing Radical Expressions

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Chapter 6 Cheat Sheet

Combining Radical Expressions: Sums and Differences

Use the Distributive Property:

an x  bn x  (a  b)n x

Examples:


Use mystery lots and combine like radicals

Examples:

Vocab:

Conjugates:

The product of two radical conjugates is a rational number.
You can use this to rationalize a denominator.

Examples:

More Problems: Pg. 378 10-15, 17-22
6.1 – 6.3 Practice Worksheet

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Chapter 6 Cheat Sheet

Section 6.4 – Rational Exponents
m
1
Rule:   n
a        AND a n  n am  (n a ) m
an

Examples:



Things to Consider when simplifying radicals:
Remember: Try and simplify radical first