Download Lecture Notes for Section 8.1 (Evaluating Roots)

Elementary Algebra Notes
Section 8.1
Page 1 of 5
Section 8.1: Evaluating Roots
Big Idea for section 8.1: Roots are the inverse procedure to taking powers, just as subtracting is the inverse
procedure to adding and dividing is the inverse procedure to multiplication.
Compare solving three types of equations:
x+3=7
x = 4 because 4 + 3 = 7
Get this answer by subtracting 3
from 7:
x+3=7
x+3–3=7–3
x=7–3
x=4
2x = 8
x = 4 because 2  4 = 8
Get this answer by dividing 8 by 4:
2x  8
2x  2  8  2
x 82
x4
x2 = 16
x = 4 because 42 = 16
Get this answer by taking the
square root of 16:
x 2  16
x 2  16
x  16
x4
ROOTS “UNDO” POWERS!
Vocabulary:
1. radical
2. radical sign
3. radicand
4. index (or order)
5. radical expression
Square Roots:
If a is a positive real number, then
a is the principal square root of a, and
- a is the negative square root of a.
For nonnegative a,
a a =a
and (- a )  (- a ) = a.
If a is not a perfect square, then a is irrational.
If a is negative, then a is not a real number.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.1
Page 2 of 5
Practice:
1. Find 169 .
2. Find  225 .
3. Find
25
.
64
4. Find an approximation of 17 .
5. Find an approximation of  42 .
Application of Square Roots: the Finding the Side of a Square Given Its Area
Practice:
6. A square has an area of 300 square feet. Find the length of each side.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.1
Page 3 of 5
Application of Square Roots: the Pythagorean Theorem
Pythagorean Theorem:
a 2  b2  c2
Practice:
7. Find the hypotenuse of a right triangle with sides of length 3” and 4”.
8. Find the approximate hypotenuse of a right triangle with sides of length 5.2” and 7.9”.
9. Find the approximate length of the leg of a right triangle with a side of length 12.95” and a hypotenuse
of length 17.81”.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.1
Page 4 of 5
Application of Square Roots: the Distance Formula
Distance Formula:
d
 x2  x1    y2  y1 
2
2
Practice:
10. Find the distance between the points (2, 7) and (-3, 5).
Higher order roots:
Definition: The principal nth root of a number
n
a , where n is an integer greater than or equal to 2, computes to a number b such that if
n
a  b , then b n  a .
If n is an even number bigger than 2, then a and b must be positive.
If n is an odd number, then a and b can be any real number.
Practice:
1. Evaluate
3
27
2. Evaluate
4
256
3. Evaluate
3
125
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.1
Page 5 of 5
4. Evaluate 4 16
5. Evaluate
4
16
6. Evaluate 3 17 to three decimal places of precision.
7. Evaluate 5 12 to three decimal places of precision.
n
2
3
4
5
6
7
8
9
10
2^n
4
8
16
32
64
128
256
512
1,024
3^n
4^n
5^n
6^n
7^n
8^n
9^n
10^n
9
16
25
36
49
64
81
100
27
64
125
216
343
512
729
1,000
81
256
625
1,296
2,401
4,096
6,561
10,000
243
1,024
3,125
7,776
16,807
32,768
59,049
100,000
729
4,096
15,625
46,656
117,649
262,144
531,441 1,000,000
2,187
16,384
78,125
279,936
823,543 2,097,152 4,782,969
6,561
65,536
390,625 1,679,616 5,764,801
19,683
262,144 1,953,125
59,049 1,048,576 9,765,625
n
2
3
4
5
6
7
8
9
10
(-2)^n
4
-8
16
-32
64
-128
256
-512
1,024
(-3)^n
(-4)^n
(-5)^n
(-6)^n
(-7)^n
(-8)^n
(-9)^n
(-10)^n
9
16
25
36
49
64
81
100
-27
-64
-125
-216
-343
-512
-729
-1,000
81
256
625
1,296
2,401
4,096
6,561
10,000
-243
-1,024
-3,125
-7,776
-16,807
-32,768
-59,049 -100,000
729
4,096
15,625
46,656
117,649
262,144
531,441 1,000,000
-2,187
-16,384
-78,125 -279,936 -823,543 -2,097,152 -4,782,969
6,561
65,536
390,625 1,679,616 5,764,801
-19,683 -262,144 -1,953,125
59,049 1,048,576 9,765,625
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.