Download Divisibility Rules

Welcome to the
Exponential Jungle
Warm Ups
• Find the Quotient: 147  3 =
• Find the Quotient: 273/3 =
• Find the Quotient: 450/10 =
• Find the Product: 4 • 4 • 4 =
• Find the Product: (-2)(-2)(-2) =
• Find the Product: (10)(10)(10)(10) =
Section 4.1: Divisibility and Factors
• One integer is DIVISIBLE by another if the
remainder is 0 when you divide.
• Divisibility Rules for 2, 5, and 10.
An integer is divisible by…
• 2, if it ends in 0, 2, 4, 6, and 8.
• 5, if it ends in 0 or 5.
• 10, if it ends in 0.
Examples: Divisibility Rules
• Is the first number divisible by the second?
• 567 and 2?
• 1,015 and 5?
• 111,120 and 10?
• 53 and 2?
• 1,118 and 2?
Divisibility Rule for 3 and 9.
• See if you can pick out the pattern…
Number
Sum of Digits
Is the Sum
Divisible by:
Is the Number
Divisible by:
3?
3?
9?
9?
282
2 + 8 + 2 = 12
Yes
No
Yes
No
468
4 + 6 + 8 = 18
Yes
Yes
Yes
Yes
215
2+1+5=8
No
No
No
No
1,017
1+0+1+7=9
Yes
Yes
Yes
Yes
Divisibility Rule for 3 and 9.
• From the pattern we saw before…
• An integer is divisible by…
• 3, if the sum of its digits is divisible by 3.
• 9, if the sum of its digits is divisible by 9.
Example Problems:
• Is the first number divisible by the
second? Explain.
• 64 by 9?
• 472 by 3?
• 174 by 3?
• 43,542 by 9?
Finding Factors
number is a FACTOR
• One integer
of another integer
number if it
divides that integer
number with
remainder zero.
Finding Factors
• So find all the different combinations of
factors
• 12? 1(12), 2(6), 3(4): so, 1, 2, 3, 4, 6, and 12
• 16? 1(16), 2(8), 4(4): so, 1, 2, 4, and 16
• 18? 1(18), 2(9), 3(6): so, 1, 2, 3, 6, 9, and 18
• 24? 1(24), 2(12), 3(8), 4(6): so, 1, 2, 3, 4, 6, 8,
12, and 24
Word Problem:
• There are 20 choral students singing at a
school concert. Each row of singers much
have the same number of students. If
there are at least 5 students in each row,
what are all the possible arrangements?
List the positive factors of each
number.
• 10
1, 2, 5, 10
• 21
1, 3, 7, 21
• 31
1, 31
Section 4.2: Exponents
• You can use EXPONENTS to show repeated
multiplication.
• (Just like multiplication can show
repeated addition).
• A POWER has two parts:
• A BASE: the factor.
• And an EXPONENT: the number of times
the base (or factor) is multiplied by
itself.
Exponents: Exponential Notation
• So show me the EXPANDED version of 2 to
the 6th power. Or 26.
• 2 • 2 • 2 • 2 • 2 • 2 = 64 = 26
Exponents…Show me!
1. Exponential Notation.
2. And Value. (The answer).
• Twelve to the first power.
• Six to the second power, or six squared.
• The opposite of the quantity seven to
the fourth power.
• Negative eight to the fifth power.
Exponential Notation and
Answers
• Twelve to the first power.
121, 12
• Six to the second power, or six squared.
62, 6 • 6 = 36
• The opposite of the quantity seven to the fourth
power.
-74, - (7 • 7 • 7 • 7)4 = - (2,401) = -2,401
• Negative eight to the fifth power.
(-8)5, (-8)(-8)(-8)(-8)(-8) = -32,768
Writing Exponential Notation
• Remember to include the Negative Sign.
• (-5)(-5)(-5) = (-5)3
• Rewrite the expression using the
commutative and associative properties.
• -2 • a • b • a • a =
• -2 • a • a • a • b = -2a3 b
Exponents and Negative Integers
• So when you multiply 2 negative
numbers…what do you get?
• A positive number
• When you multiply 3 negative
numbers…what do you get?
• A negative number
Exponents and Negative Numbers
• When you multiply an EVEN number of
negative integers, the answer will be
positive.
• When you multiply an ODD number of
negative integers, the answer will be
negative.
Example Problems
• Write these in exponential notation…
• 6•6•6•6=
• (-3)(-3)(-3) =
• 4 • y • x • y = 4y2x
Word Problem:
• A microscope can magnify a specimen 103
times. How many times is that?
Example Problems:
• Simplify: 62
• Evaluate: - a4, for a = 2
• Evaluate: (-a)4, for a = 2
Orders of Operations
1. Work inside the grouping symbols
2. Simplify any terms with exponents.
3. Multiply and divide in order from left to
right.
4. Add and subtract in order from left to
right.
Example Problem:
•
4(3+2)2 =
1. Grouping Symbols: 3+2 = 5
2. Exponents: 52 = 5 • 5 = 25
3. Multiplying/Dividing: 4 • 25 = 100.
4. Addition/Subtraction?: Nope
•
So, 100 is the answer.
Example Problem:
•
-2x3 + 4y, for x = -2 and y = 3.
•
Substitute Variables: -2(-2)3 + 4(3)
1. Grouping Symbols: None
2. Exponents: (-2)3 = (-2)(-2)(-2) = -8
-2(-8)+ 4(3)
3. Multiplication/Division: 16 + 12 = 28
4. Addition/Subtraction: None
•
So, 28 is the answer.