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Mastering the Art of Simplifying Fractions: Prime Factorization and Composite Numbers

In this section, we explore the process of simplifying fractions, focusing on determining the prime factorization of numbers and identifying prime versus composite numbers. A prime number only has factors of 1 and itself, while composite numbers include factors beyond 1. We’ll discuss effective methods for simplifying fractions, including using prime factorization and the greatest common factor (GCF). Learn how to apply these methods in practical examples and how to handle fractions with variables.

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Mastering the Art of Simplifying Fractions: Prime Factorization and Composite Numbers

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  1. Section 4.2 Section 4.2: Simplifying Fractions

  2. Section 4.2: Simplifying Fractions • Determining the prime factorization of a number • Simplifying fractions to lowest terms

  3. Prime versus Composite Numbers A prime number is a number whose only factors are 1 and the number itself. 2 , 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … A composite number is any number which is not prime 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, … The number 1 is considered neither prime nor composite.

  4. State whether the number is prime composite or neither. 39 • Prime • Composite • Neither

  5. State whether the number is prime composite or neither. 53 • Prime • Composite • Neither

  6. Prime Factorization • One method to factor a whole number is to make a factor tree. Begin by determining any two numbers that when multiplied equal 220. Then continue factoring each factor until the branches “end” in prime numbers. Therefore, the prime factorization of 220 is TRY: 90, 126, 78

  7. Simplifying Fractions Method 1: Using Prime Factorization Method 2: Dividing by the Greatest Common Factor (GCF)

  8. Method 1: Simplifying with Prime Factors Process: • Find the prime factorization for both the numerator and denominator • Cross out those factors that are in both the numerator and the denominator. • Note: Only cross out common factors on a 1-to-1 basis

  9. Method 2: Simplify with GCF Process: • Identify the Greatest Common Factor (GCF) or a number that divides into both the numerator and denominator • Divide both the numerator and denominator by the GCF Example: Try: , - , ,

  10. Simplifying Fractions with the TI-34 To simplify 27/36 using the TI-34 do the following; Enter 27 Press the button Enter 36 Press [enter] Press the [simp] button and then [enter] Fac = 3 so we need to simplify again Press the [simp] button then [enter] Fac = 3 so we need to simplify again Press the [simp] button and then [enter] Fac = 1 so we can stop, the answer is 3/4

  11. Simplify Fractions with Variables Process: • Factor the numerator and denominator • Eliminate common factors Examples: = = = = - = - = -

  12. Try These: • -

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