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Ch 8 – Powers and Roots
8.1 – Powers and Exponents
Perfect Square:
Exponent:
Powers:
Example: Write each expression using exponents.
5·5·5
d·d·d·d·d·d·d
p
Example: Write (6)(6)(-7)(-7)(-7)(-7)(-7) using exponents.
Example: Write each power as a multiplication expression.
64
h5
7a3b2
Example: For every 3-digit telephone prefix, the number of possible 4-digit telephone extensions is 104.
Write this number as a multiplication expression and then evaluate the expression.
Order of Operations:
1.
2.
3.
4.
Example: Evaluate each expression.
5a2 if a = 4
3p2-q3 if p = -4 and q = -1
-5(m + n)2 if m = 4 and n = 2
8.2 – Multiplying and Dividing Powers
Product of Powers:
Example: Simplify each expression.
52 · 57
s10 · s5
(10a)(5a2)
(m5n4)(m3 n7)
Quotient of Powers:
Example: Simplify each expression
76
72
p10
p
15a 6b 4
3a 4b3
Divide a power by itself:
Example: Simplify
12c 5 d 7
4c 2d 4
10 x 4 y 3
5x 4 y 2
8.3 – Negative Exponents
Negative Exponents:
Example: Write 6-2 using positive exponents. Then evaluate the expression.
Example: Write 2-4 using positive exponents. Then evaluate the expression.
Example: Simplify each expression.
q 3r 4
m 2 n10
m5 n 2
 6r 3 s 5
18r  7 s 5t  2
 10h 6 k 4
25h 3 k 7
5a 4b 6
 25a  2b 6c 3
Example: A large archery target has a diameter of 1 meter. An arrow tip has a diameter of 10-2 meter. How
many arrows could fit across the diameter of the target?
Power of a Power:
8.4 – Scientific Notation
Prefix
Power of 10
Metric Prefixes
Meaning
Prefix
Power of 10
Meaning
Metric system:
Multiplying by Powers of 10:
Examples: Express each measurement in standard form.
8 kilobytes
2.5 microseconds
2 gigabytes
Scientific Notation:
Steps to write a number in scientific notation:
Example: Express each number in scientific notation.
325,000
0.00028
3,900,000,000
Example: Evaluate each expression.
30 x 30,000,000
2000 x 3,000,000,000
5.4 108
2 105
Example: A light ray from the sun travels about 186,000 miles per second. How far does it travel in 5 x 102
seconds? Use the formula d = rt, where r is the speed of light and t is the time in seconds.
8.5 – Square Roots
Square Root:
Example: Simplify each expression.
100
 81
121
Radical Expression:
Prime Number:
Composite Number:
Simplifying Radical Expressions:
Prime Factorization:
Product Property of Square Roots:
Example: Simplify each expression
256
400
Quotient Property of Square Roots:
Example: Simplify each expression

100
9
64
81
Example: The formula r  0.56 A gives the approximate radius of r of a circle that has an area of A square
units. If a circle has an area of 15 square centimeters, find the radius of the circle to the nearest tenth of a
centimeter.
8.6 – Estimating Square Roots
Irrational Numbers:
Example: Estimate each square root
48
45
200
190
8.7 – Pythagorean Theorem
Right Triangle:
Pythagorean Theorem:
Example: Find the length of the hypotenuse of the right triangle.
Example: Find the length of the hypotenuse of a right triangle if the lengths of the legs are 6 meters and 8
meters.
Example: Find the length of one leg of a right triangle if the length of the hypotenuse is 22 centimeters and
the length of the other leg is 15 centimeters. Round to the nearest tenth.
Example: Find the missing measure.
Converse:
Converse of the Pythagorean Theorem:
Example: The measures of the three sides of a triangle are 6, 11, and 13. Determine whether this triangle is
a right triangle.
Example: The measures of the three sides of a triangle are 6, 8, and 10. Determine whether this triangle is a
right triangle.
Example: When laying out a rectangular driveway, a concrete worker measures along one side a distance of
16 feet and along the adjacent side a distance of 30 feet. The measure of the hypotenuse is 34 feet. Is the
corner of the driveway square?