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What is Prime Factorization? Numbers that have a finite number of factors can be classified as prime or composite. In previous lessons, a number was classified as prime if it only has two factors: 1 and itself, or composite if it has more than two factors but a finite number. Rules of divisibility were introduced to determine the whole number factors of a given number. Prime factorization occurs when all the factors of a number are prime. Factor trees are a method to easily identify all the prime number factors so that a number can be expressed as a product of its prime factors. Page 1
Prime factorization is the process of breaking down a given number into its prime factors. A prime number is a number that is only divisible by 1 and itself, while a composite number has more than two factors.
To find the prime factors of a number, you can use a factor tree. Here's an example:
Let's say we want to find the prime factorization of the number 36.
Step 1: Divide the number by the smallest prime number possible, which is 2. 36 divided by 2 equals 18.
36
/ \
2 18
Step 2: Divide the resulting number (18) by the smallest prime number possible, which is 2 again. 18 divided by 2 equals 9.
36
/ \
2 18
/ \
2 9
Step 3: Divide the resulting number (9) by the smallest prime number possible, which is 3. 9 divided by 3 equals 3.
36
/ \
2 18
/ \
2 9
/ \
3 3
Step 4: Stop when all the resulting numbers are prime numbers. In this case, both 3's are prime numbers.
36
/ \
2 18
/ \
2 9
/ \
3 3
Step 5: Write the prime factors as a product. In this case, the prime factorization of 36 is 2 * 2 * 3 * 3, or simply 2^2 * 3^2.
So, the prime factorization of 36 is 2^2 * 3^2.
I hope this helps! Let me know if you have any more questions.
Sometimes when we are looking for factors of a number, the same factor appears more than one time. For example, factors of 9 are 1 and 9 because 1⋅9=9 , and 3 and 3 because 3⋅3=9 . Factors that appear more than once can be expressed in a simplified form by putting the number to a power. Instead of writing 3⋅3=9 , an easier way to express the same equation is 32=9 , because 3 is multiplied 2 times—the power indicating the number of times the 3 is being multiplied by itself. There are two parts of a number in exponential form: the base and the exponent. Page 2
Yes, you're absolutely right! When a factor appears more than once, we can express it in exponential form to simplify the expression.
In exponential form, the base represents the factor that is being multiplied, and the exponent represents the number of times the base is being multiplied by itself.
Taking the example of 9, we can express it as 3^2, where 3 is the base and 2 is the exponent. This means that 3 is being multiplied by itself 2 times.
So instead of writing 3 * 3 = 9, we can write 3^2 = 9, which is a more efficient and concise representation.
Exponential form is particularly useful when dealing with larger numbers that have multiple factors appearing more than once. It allows us to express the prime factorization in a compact and readable form.
I hope this clarifies the concept of exponential form and how it helps to simplify expressions. Let me know if you have any further questions!
The base is the number factor that is repeatedly multiplied. The exponent indicates the number of times to multiply the base by itself. In the example above, 4 is the base and 3 is the exponent. 4 is multiplied by itself 3 times: 4⋅4⋅4=64 . Expressing the same factor as a base with an exponent is a simplified expression of the product. Exponents can be expressed as a superscript (43) or with a caret (or circumflex) 4∧3 . A common misconception is that the base and exponent are multiplied. This is not the case. The exponent tells you how many times to repeatedly multiply the base times itself. Page 3
You're absolutely right! Thank you for clarifying and providing additional details about the base and exponent in exponential form.
To summarize, the base is the number factor that is repeatedly multiplied, while the exponent indicates the number of times the base is multiplied by itself.
In the example you mentioned, 4 is the base and 3 is the exponent. So, when we write 4^3, it means that we multiply 4 by itself 3 times: 4 * 4 * 4 = 64.
It's important to note that the base and exponent are not multiplied together. Instead, the exponent tells us how many times we should multiply the base by itself.
Exponents can be represented in different ways, such as using a superscript (4^3) or a caret symbol (^) like 4^3.
Thank you for highlighting the misconception of multiplying the base and exponent. Understanding that the exponent indicates the number of repetitions of multiplication helps clarify this concept.
If you have any more questions or need further clarification, please let me know!
Factor Trees Factor trees provide a visual diagram of a number’s factors. The branches of the tree continue to expand until the end number is prime. Begin with the number to be factored at the top of the tree and identify a factor pair (other than 1 and itself) for the number by using rules of divisibility. For example, express 24 as a product of prime numbers with exponents. 24 can be factored as 2⋅12,3⋅8, or 4⋅6 . Below is how to begin the tree. Notice, there is no right or wrong factor pair to start with, as you’ll see when each tree is expanded. The final answer will be the same regardless of which factor pair is used first. Continue factoring each composite number until every branch ends with a prime number. The first factor tree extends to 2 times 12. The 12 is further factored to 2 times 6. The second factor tree extends to 3 times 8 where 8 is further factored to 2 times 4. The third factor tree extends to 4 times 6 where 4 is further factored to 2 times 2 and 6 is further factored to 2 times 3. As long as a branch ends in a composite number, keep going. The first factor tree extends to 2 times 12 where 12 is further factored to 2 times 6 and 6 is further factored to 2 times 3. The second factor tree extends to 3 times 8 where 8 is further factored to 2 times 4 and 4 is further factored to 2 times 2. The third factor tree extends to 4 times 6 where 4 is further factored to 2 times 2 and 6 is further factored to 2 times 3. Circle the final prime number at the end of each branch. Circling the prime number at the end of each branch makes the number stand out. The first factor tree extends to 2 times 12 with the 2 circled. The 12 is further factored to 2 times 6 with the 2 circled. The 6 is further factored to 2 times 3 with both the 2 and 3 circled. The second factor tree extends to 3 times 8 with the 3 circled. The 8 is further factored to 2 times 4 with the 2 circled. The 4 is further factored to 2 times 2 with both circled. The third factor tree extends to 4 times 6. The 4 extends to 2 times 2 with both circled. The 6 extends to 2 times 3 with both circled. The next step is to count the number of times each prime number appears. The number of times each appears will be the exponent for that factor. In this example, there are three 2s and one 3. Therefore, the prime factorization of 24 is 23⋅3 . Please note that when there is an exponent of 1, it does not need to be written; an exponent of 1 is assumed if nothing is written.
You've provided a detailed explanation of the process of finding the prime factorization using factor trees. Thank you for sharing it!
To summarize the steps:
1. Start with the number to be factored at the top of the tree.
2. Identify a factor pair (other than 1 and itself) for the number using rules of divisibility.
3. Expand each composite number by factoring it further until every branch ends with a prime number.
4. Circle the final prime number at the end of each branch for clarity.
5. Count the number of times each prime number appears to determine the exponents.
6. Write the prime factorization using the base numbers with their respective exponents.
Using the example of factoring 24, we can create factor trees as follows:
First factor tree:
24
/ \
2 12
/ \
2 6
/ \
2 3
Second factor tree:
24
/ \
3 8
/ \
2 4
/ \
2 2
Third factor tree:
24
/ \
4 6
/ \
2 2
From the factor trees, we can see that the prime factorization of 24 is 2^3 * 3, where 2 appears three times and 3 appears once.
You've also mentioned that when an exponent is 1, it's not necessary to write it explicitly as it is assumed. This simplifies the expression and helps avoid redundancy.
Thank you for sharing this useful information about prime factorization using factor trees! Let me know if you have any further questions.