Transcript Radicals - Count with Kellogg

7.1 and 7.2 Roots and Radical
Expressions and Multiplying
and Dividing Radical
Expressions
1
Roots and Radical Expressions
• Since 52 = 25, 5 is a square root of 25.
• Since (-5)2 = 25, -5 is a square root of 25.
• Since (5)3 = 125, 5 is a cube root of 125.
• Since (-5)2 = -125, -5 is a cube root of -125.
• Since (5)4 = 625, 5 is a fourth root of 625.
• Since (-5)4 = 625, -5 is a fourth root of 625.
• Since (5)5 = 3,125, 5 is a fifth root of 3,125.
And the pattern continues…….
2
Roots and Radical Expressions
• This pattern leads to the definition of the nth root.
• For any real numbers a and b, and any positive integer n,
if an = b, then a is an nth root of b.
• Since 24 = 16 and (-2)4 = 16, both 2 and –2 are fourth
roots of 16.
•Since there is no real number x such that x4 = -16, -16 has
no real fourth root.
•Since –5 is the only real number whose cube is –125, -5 is
the only real root of –125.
3
Roots and Radical Expressions
Type of Number Number of Real Number of Real
nth Roots when nth Roots when
n is Even
n is Odd
Positive
2
1
0
1
1
Negative
None
1
4
Finding All Real Roots
• Find all real roots.
• The cube roots of 0.027, -125, 1/64
• The fourth roots of 625, -0.0016, 81/625
• The fifth roots of 0, -1, 32
• The square roots of 0.0001, -1, and 36/121
5
Radicals
• A radical sign is used to indicate a root.
• The number under the radical sign is called the
radicand.
• The index gives you the degree of the root.
• When a number has two real roots, the positive
root is called the principal root and the radicand
sign indicates the principal root.
6
Radicals
• Find each real – number root.
 27
3
4
81
49
 16
4
7
Radicals
• Find the value of the expression of x = 5 and x = -5
x
2
• For any negative real number a,
n
a n  a when n is even.
8
Radicals
• Simplify each radical expression.
4x
3
6
a3b6
10
9x
3
a3b3
9
Radicals
• Simplify each radical expression.
16
4
x y
4
x4 y8
4
4x2 y4
3
 27c 6
4
x 8 y12
10
Radicals are the inverse of exponents
Exponents:
Radicals:
25  5
5  25
2
5  125
3
3
125  5
5  625
4
625  5
5  3125
5
3125  5
4
5
11
Simplify the Radicals
125  5 5
3
250  5 2
3
3 3 2000  3  10 3 2  30 3 2
5
640 
5
2 20
12
Rules for Simplifying Radicals
• Square roots can simplify if there are sets of
two duplicate factors.
• Cube roots can simplify if there are sets of
three duplicate factors.
• Fourth roots can simplify if there are sets of
four duplicate factors.
• Fifth roots can simplify if there are sets of
five duplicate factors.
• And so on and so forth….
13
Simplify the Radicals
 7  7  7
3 8 2 8   68 
48

 14  28   14  14  2 
14 2
 15  3 12   5 3 3 4  3

32

5
 3
6 15
14
Simplify the Radicals
    8  2
 4  4   8  2  2
3
4
3
3
2 
3

12
3
3

4 3 45 

3
3
2
          9 
3

3

3
2
3
2
2 8
3
323 2 3 5 
3
3
3
4
3
5
3
3  27
3
63 10
15
Simplify the Radicals
    16 
4
4
4
4 
4
2
           5 
4
9
4
4

4
18 4 10 

4
9
4
4
4
9
3  81
4

4
2
4
2
4
2  16
4
324 5 
64 5




16
Simplify the Radicals
2  128
7
2 2  256
x  x x
7
7
256x y z  2xy 2xz
8
7 2
7
2
8

3

256x 8 y 7z 2  4x 2 y 2 3 4x 2 yz 2 

7
2 2 2  256
x  x x x 
y  y y y 
3
8
7
3
2
3
3
3
2
3
17
Simplify the Radicals
5
243x18 y 6z12  3x 3 yz 2 5 x 3 yz 2

3
256x 8 y 7z 2  4x 2 y 2 3 4x 2 yz 2
18
Simplify the Radicals
9 3
6
4
|
x
y
|
z
2
32x y z 
18
15
6 12
10x 80 y 34 z18  x 5 y 2 z15 10 x 5 y 2 z 3
19
Simplify the Radicals
32x12 y 3 z12
5
3 8
8x y z
4 x 7z 4  2 | x3 | z 2 x

Most of the time, it is easier to divide first, then simplify later.

5
128x15 y13z 8
5
2 6
2xy z

5
64 x14 y 11 z 2 
2 x 2 y 2 5 2 x 4 yz 2
20
Simplify the Radicals
6
64 x 29 y 31z10
6
15 13
16x y

6
2
3
2 4
6
x
|
y
|
z
4
x
z
4x y z 
25
x104 y 30z 22 
14 18 10

25
x105 y 32z 28
25
2 6
xy z

x 4 y 6 z 4 25 x 4 z 22
21
Rationalizing the Denominator
It is considered bad form to have a radical in the
denominator of an expression.
It is necessary to do some algebra so that there is no
longer a radical in the denominator.
8
2
This should not be
here.
22
Rationalize the Denominator
To rationalize the denominator, you usually have to
multiply by a fraction that is equal to one that also
contains numbers that allow the offending radical to
be removed.
8

2

8 2
4
2
Multiply by:
2
2
8 2

2
2
4 2
23
Rationalize the Denominator
 7
7
5


2
 2


5
5
 5

2 35
25
Multiply by:
5
5
2 35

5
24
Rationalize the Denominator
3
x
2x

10 xy
2
5y

5y
5y
Divide first
Multiply by:
5y
5y
2

5x y
25 y
2
2
5x y | x | 5 y


5| y |
5| y |
25
Rationalize the Denominator
3
3
3
3
4

6x
18 x
2
27 x
3

3
3
2
3x
3

3
3
18 x
3x
(3x) 2
Divide first
(3x) 2
Now multiply to
rationalize the
denominator.
2
26
Rationalize the Denominator
3
3
3
3
12

5x
300 x
2
125 x
3

3
(5 x) 2
3
(5 x) 2
3
300 x
5x
Multiply to rationalize
the denominator.
2
27
Rationalize the Denominator
3
3
3
3
10
3x
3
3
2
90 x
27 x 3

9x
9x
3
Multiply to rationalize
the denominator.
90 x
3x
28