Download 4-1 Factors and Monomials

grade7mathchap4
4-1 Factors and Monomials
The factors of a whole number divide that number with a remainder of 0. For example, 2
is a factor of 12 because 12 ÷2 = 6, and 5 is not a factor of 12 because 12 ÷ 5 = 2 with a
remainder of 2.
Another way of saying that 6 is a factor of 12 is to say that 12 is divisible by 6.
Sometimes you can test for divisibility mentally. These rules will help you determine
whether a number is divisible by 2, 3, 5, 6, or 10.
A number is divisible by:
2 if the ones digit is divisible by 2
3 if the sum of its digits is divisible by 3
5 if the ones digit is 0 or 5
6 if the number is divisible by 2 and 3
10 if the ones digit is 0
An expression like 5x is called a monomial. A monomial is an integer, a variable, or a
product of integers or variables. Other monomials are z, 16, mn, and 6ab. Expressions
like 4x-2 and x + 3 are not monomials since they involve addition or subtraction.
4-2 Powers and Exponents
The expression 6 x 6 can be written in a shorter way using exponents. An exponent tells
how many times a number, called the base, is used as a factor. Numbers that are
expressed using exponents are called powers.
The expression 6 x 6 can be written as 62.
72 is seven to the second power or seven squared
63 is six to the third power or six cubed
104 is ten to the fourth power
(-2)5 is negative two to the fifth power
Any number, except 0, raised to the zero power, like 50, is defined to be 1.
The number 12, 496 is in standard form. You can use exponents to express a number
in expanded form. 1 x 10,000 + 2 x 1000 + 4 x 100 + 9 x 10 + 6
Order of Operations
1. Do all operations within grouping symbols first; start with the innermost
grouping symbols.
2. Evaluate all powers in order from left to right.
3. Next do all multiplications and divisions in order from left to right.
4. Then do all additions and subtractions in order from left to right.
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grade7mathchap4
4-3 Problem-Solving Strategy: Draw a Diagram
When we are able to draw a diagram for a given situation, we often are able to understand
it better. Drawing a diagram is a powerful problem-solving strategy.
4-4 Prime Factorization
A prime number is a whole number greater than one that has exactly two factors, 1 and
itself.
Numbers of squares like 4,, 6, 8, 9, and 10 that can be arranged in rectangles other than a
row or a column are called composite numbers. A composite number is a whole number
greater than one that has more than two factors. A composite number can always be
expressed as a product of two or more primes.
The numbers 0 and 1 are considered neither prime nor composite.
Every number is a factor of 0, since any number multiplied by zero is zero.
The number 1 has only one factor, itself.
Every whole number greater than 1 is either prime or composite.
A factor tree is a diagram used to find prime factors. The factoring process ends when
all of the factors are prime. When a positive integer (other than 1) is expressed as a
product of factors that are all prime, the expression is called the prime factorization.
The prime factorization of a negative integer ( other than -1) contains a factor of -1, and
the rest of the factors are prime.
You can also use a strategy called the cake method to find a prime factorization. The
cake method uses division to find factors.
Begin with the smallest prime that is a factor.
Then divide the quotient by the smallest possible prime factor.
Repeat until the quotient is prime.
A monomial can also be written in factored form as a product of prime numbers, -1, and
variables with no exponent greater than 1.
4-5 Greatest Common Factor
The greatest of the factors of two or more numbers is called the greatest common factor.
There are many methods for finding the greatest common factor. One method is to
simply list the factors of each number and identify the greatest of the factors common to
the numbers.
Another method for finding the greatest common factor of two or more numbers is to find
the prime factorization of the numbers and then find the product of their common factors.
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4-6 Simplifying Fractions
A ratio is a comparison of two numbers by division. A ratio can be expressed in several
ways. The expressions below all represent the same ratio.
2 to 3
2:3
2
3
2÷3
A ratio is most often written as a fraction in simplest form. A fraction is in simplest
form when the GCF of the numerator and the denominator is 1.
4-7 Using the Least Common Multiple (LCM)
A multiple of a number is a product of that number and any whole number.
Multiples that are shared by two or more numbers are called common multiples.
The least of the nonzero common multiples of two or more numbers is called the least
common multiple (LCM) of the numbers.
The LCM of two or more monomials is found in the same way as the LCM of two or
more numbers.
One way to compare fractions is to write them with the same denominator. While any
common denominator could be used, the least common multiple of the denominators, or
the least common denominator (LCD) is usually the most convenient.
4-8 Multiplying and Dividing Monomials
Products of Powers
in words: You can multiply powers that have the same base
by adding their exponents.
in symbols:
Quotient of Powers
For any number a and positive integers m and n,
am ∙ an = am+n.
in words:
You can divide powers that have the same base
by subtracting their exponents.
in symbols:
For any nonzero number a and whole numbers
m and n, am= am-n
an
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4-9 Negative Exponents
You can use the quotient of powers rule and the definition of a power to simplify the
expression x3 and write a general rule about negative powers.
x6
Negative Exponents
For any nonzero number a and any integer n a-n = 1
an
You can use the prime factorization of a number to writ a fraction as an expression with
negative exponents. Expressions involving variables can also contain negative
exponents.
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