Download a0 = 1

Ch. 6 Notes
6.1: Roots and Radical Expressions
Review: Properties of Exponents
a0 = 1
a-n =
(am)n = amn
(ab)n = anbn
am(an) = am+n
Every power has a root. Squares have a square root. Cubes have a cube root.
22 = 4
23 = 8
24 = 16
25 = 32
2 is a square root of 4
2 is a cube root of 8
2 is the fourth root of 16
2 is the 5th root of 32
Study the vocabulary and figure to the right.
Examples: Find all the real square roots of each number.
1. 400
2. 196
3. 10,000
Examples: Find all the real cube roots of each number.
5. 216
6. 343
7. 0.064
8.
Examples: Find all the real fourth roots of each number.
9. 81
10. 256
Examples: Find each real root.
12.  25
13. 0.01
14.
4
0.0081
1000
27
11. 0.0001
15.
3
27
16.
3
27
What is the answer to
?
What is the answer to
?
What is the answer to
?
Examples: Simplify each radical expression. Use absolute value symbols when needed.
17.
121y10
18.
3
8g 6
20.
25( x  2)4
21.
3
0.008
Examples: Find the 2 real roots of the equation.
22)
19.
5
243x5 y15
6.2: Multiplying and Dividing Radical Expressions
What are the perfect squares greater than 1?
Examples: Write true or false for the following.
You are allowed to combine radicals that have the same index only!
Similar to combining like terms!
Simplify the new radicand and find the solution then!
When going from one radicand to multiple ones, find all the necessary roots before splitting the
radicals.
Examples: Multiply, if possible. Then simplify.
1. 3 45  3 75
2. 18  50
3.
3
16  3 4
4
256s 7 t12
Examples: Simplify. Assume that all variables are positive.
4.
3
16a12
x2 y10 z
5.
6.
Examples: Multiply and simplify. Assume that all variables are positive.
7.
9 x2  9 y5
8.
3
50 x2 z 5  3 15 y3 z
9. 4 2 x  3 8 x
When doing division, same rules apply as multiplication.
Indexes the same, means you can combine into one and try to simplify.
Rationalizing the Denominator: radicals are not allowed in the denominator, you must multiply
top and bottom by the radical to remove it
You only need to worry about rationalizing when the fraction won’t reduce all the way
Examples: Divide and simplify. Assume that all variables are positive.
75
10.
3
13.
4
11.
54 x 5 y 3
2x2 y
243k 3
3k 7
3
12.
14.
4x 2
3
x
(2 x ) 2
(5 y ) 4
Examples: Rationalize the denominator of each. Assume all variables are positive.
15.
y
5
16.
18 x 2 y
2 y3
3
17.
3
7 xy 2
4 x2
6.3: Binomial Radical Expressions
Like Radicals:
Using properties of real numbers, you are allowed to combine like radicals.
Simplify all radicals before trying to combine like ones.
Remember that the radicand does not change, only the coefficient. Similar to combining like
terms.
Adding and Subtracting:
Simplify all radicals. Then combine coefficients of like radicals.
Multiplying Binomials:
Use FOIL method and combine like radicals.
Examples: Simplify if possible.
1. 9 3  2 3
2. 3 7  7 3 x
3. 14 3 xy  3 3 xy
5. 7 x  x 7
6. 3 32  2 50
7.
9. (1  5)(2  5)
2
10. ( 2  7)
11. (3 2  9)(3 2  9)
3
81  3 3 3
4.
3x  2 3x
8. 2 4 48  3 4 243
12. ( 11  5)( 11  5)
Conjugates: Radical expressions that only differ in signs as below
11  5
11  5
Examples: Rationalize each denominator. Simplify the answer.
13.
3  10
5 2
14.
2  14
7 2
15.
2 3 x
3
x
6.4: Rational Exponents
Numbers that have fractions for exponents can be re-written as radicals.
The denominator of the fraction is always the index.
The base is the radicand and the numerator is the new power of the base or radicand.
When trying to simplify, always write in prime factorization.
You must be able to write in radical form and then back to rational exponent form.
Examples: Write each expression in radical form.
4
1.5
1. x 3
2.
a
3. M2.4
Examples: Write each expression in exponential form.
4.
x3
5.
3
m
6.
3
2y 2
Examples: Simplify each expression.
1
7.
125 3
1
8. 32 5
1
1
9. 8 4  32 4
When simplifying radicals, these properties along with converting from exponential to radical
form will allow you to simplify expressions.
Examples: Write each expression in simplest form. Assume that all variables are positive.
2
1
10.
(814 ) 4
13.
x2  x3
1
1
 x7
 2
3
x
4
16.
11.
2
83
1
( 27) 3
15.
( y 3 ) 9
2
2
14. (3 x 2 )(4 x 3 )



12.
1
17.
 27 x6  3

4 
 64 y 
18.
19.
1
6
x x x
1
3
2
1
1
x3 y 2
1
1
4
1
x2 y3
20.
 12 x8  2

10 
 75 y 
6.5: Solving Square Root and Other Radical Equations
Radical Equation:
Square Root Equation:
Steps for Solving:
1) Always get the radical by itself. (Nothing in the radicand can be moved in this step)
2) Undo the index by raising each side to the reciprocal of the index.
3) Solve the resulting equation.
Always go back and check your answers!
Extraneous Solutions are ones that do not check!
Examples: Solve
1. 5 x  2  12
2.
2x  5  7
3.
3x  3  6  0
4.
3
13x  1  4  0
Examples: Solve
1
5.
( x  2) 3  5
1
1
6.
2x 3  2  0
7.
(2 x  1) 3  1
Examples: Solve. Check for extraneous solutions.
8.
x 1  x 1
9.
2 x  1  3
1
10. ( x  7) 2  x  5
6.8: Graphing Radical Functions
Square root functions are inverses of quadratics.
We put restriction on the variable so that the function passes the vertical line test.
This means we don’t include the “±” in front of the radical.
Quadratics (no restrict.)
Square Root:
y
y
x
x
Quad. ( x ≥ 0 )
Square Root:
y
y
x
x
Translating the Square Root Vertically
Examples: Graph the following.
1. y 
y
x 1
x
Translating the Square Root Horizontally
Translating Horizontal and Vertical
2. y 
3. y 
x5
x 1  5
y
y
x
x
Graphs of Cubic Root Functions:
What does y 
3
What would y 
x look like?
y
3
x  5 look like?
y
x
Do we need to worry about restrictions?
Solutions found by Graphing
Graph the given function.
Find the spot where it crosses the x-axis.
I.E.----You must find the x-intercepts still!!!
x