Factorizations and Exponents

1
Factorizations and Exponents
2
Todays Learning Goals

• We will continue to develop our understanding of
  factors and factoring.

• We will introduce the meaning of exponents.

3
Factorizations

• Previously, we discussed the following
  factorizations of 12.

112 12
26 12
34 12

• Above, we have 12 as the product of only two
  factors. However, some numbers can be written as
  the product of several different factors.

• How could we write 12 as the product of three
  factors (1 cannot be one of the factors)?

Nice232 12 OR 322 12
4
Factor Trees

• To determine a factorization that has more than
  two products, it often helps if we create a
  factor tree.

• A tree is made from its roots. Similarly, a
  number is made of its factors.

• A factor tree resembles the roots of a tree like
  in the picture at the right.

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Factorizations

• Lets make a factor tree for the number 24.

• The top of the tree is the number 24.

• Next, find two factors of 24 that are different
  than 1 and itself (they are obvious factors of
  any number).

• Below, we have three different ways to start.

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Factorizations

• Where possible, break down each of the factors
  into the product of two factors (again not using
  1).

• What could we break down in the first factor tree?

Nicethe 12 can be broken down.
7
Factorizations

• What could we break down in the second factor
  tree?

Nicethe 8 can be broken down.

• What could we break down in the third factor tree?

Niceboth the 4 and 6 can be broken down.
8
Factorizations

• The first factor tree shows that 24 is the
  product of what three factors?

Yes24 2 x 6 x 2.

• The second factor tree shows that 24 is the
  product of what three factors?

Good24 3 x 4 x 2.
24
24
24
2
12
3
8
4
6
x
x
x
x
2
2
2
x
x
x
6
4
2
2
3
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Factorizations

• The third factor tree shows that 24 is the
  product of what four factors?

Great24 2 x 2 x 2 x 3.
24
24
24
2
12
3
8
4
6
x
x
x
x
2
2
2
x
x
x
6
4
2
2
3
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Factorizations

• Is it possible to break down the first factor
  tree anymore?

Excellentthe 6 can be broken down.

• Is it possible to break down the second factor
  tree anymore?

Correctthe 4 can be broken down.
24
24
24
2
12
3
8
4
6
x
x
x
x
2
2
2
x
x
x
6
4
2
2
3
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Prime Factorization

• All three factor trees show that 24 is the
  product of what four factors?

Nice2 x 2 x 2 x 3.

• When we break a number down all the way to the
  product of primes, it is called a prime
  factorization.

24
24
24
2
12
3
8
4
6
x
x
x
x
2
2
2
x
x
x
6
4
2
2
3
2
x
2
x
3
2
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Exponents

• Notice that we had 24 2 x 2 x 2 x 3. Another
  way to write 2 x 2 x 2 is 23.

• The small raised 3 in 23 is called an exponent.
  An exponent tells us how many times the factor is
  repeated.

• We read 23 as 2 to the third power or 2 cubed.

• Is 23 the same as 2 x 3?

No23 2 x 2 x 2 8 while 2 x 3 6

• What is the value of 24 x 52?

Yes24 2 x 2 x 2 x 2 16 and 52 5 x 5 25.
So, 16 x 25 400.
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Exponents

• Continue the factor tree for 100 below.

• How can we write 100 as the product of two
  factors?

Great100 10 x 10.

• How can we write 100 as the product of three
  factors?

Nice100 5 x 2 x 10.

• How can we write 100 as the product of four
  factors?

Correct100 5 x 2 x 5 x 2.

• How can we re-write the prime factorization with
  exponents?

Excellent100 22 x 52.
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Partner Work

• You have 20 minutes to work on the following
  problems with your partner.

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For those that finish early

• 1. Find the longest factorizations for 72 and
  120.
• 2. What kinds of numbers are in the longest
  factor strings for the numbers you found?
• 3. Rewrite the factor strings you found for 72
  and 120 using shortcut notation with exponents.

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Big Idea from Todays Lesson

• A number can be broken down into factors.
• 1 and the number itself are always factors.
• A factor tree can be helpful to see how to break
  down a number into smaller factors.
• A prime factorization is the longest factor
  string you can make.
• 45 means 4 x 4 x 4 x 4 x 4not 4 x 5!

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Homework

• Complete Worksheet.
• Pgs. 137 138 (25 32, 33, 34, 38)