EXPONENTS, FACTORS AND FRACTIONS • In a mathematical expression where the same number is multiplied many times, it is often useful to write the number as a base with an exponent. The exponent represents the number of times to multiply the number, or base. • When a number is represented in this way it is called a power. • Large numbers can often be rewritten as a product of prime numbers. This is called prime factorization. The number, 384, written with prime factorization is 3 · 2 ˆ7. • Exponents are also used to evaluate numbers. Any number to a zero exponent is 1 and any number to a negative exponent is a number less than 1. • Exponents are used in scientific notation to make very large or very small numbers easier to write. • Just as prime factorization uses factors, so does simplifying fractions. Simplifying fractions is the process of reducing fractions and putting them into their lowest terms. • Fractions may need to be simplified after a mathematical operation, such as multiplication, which produces a solution that is very large. • Simplifying fractions can also be used to compare and order fractions with different denominators. • Mixed numbers and improper fractions can also be simplified in order to compare with other fractions. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
How to use exponents, factors and fractions: • Prime factorization is the process of breaking a number into its prime numbers and then writing them as a product. The use of exponents is common with prime factorization. To find the prime factorization of a number, break it into two numbers that are its factors. By repeating the process, until all the numbers are prime, the result will be the product of the prime numbers. Example: What is the prime factorization of 120? 120 ۸ 10 12 ۸ ۸ 5 2 3 4 ۸ 2 2 The prime factorization of 120 is 5 · 2 · 3 · 2 · 2 = 5 · 3 · 2³ • To solve for a power, simply multiply the base the number of times indicated by the exponent. If a base has the symbol, ^, and then a number, it means the number after the ^ is the exponent. If a number is raised to zero, the answer is always 1. If a number is raised to a negative exponent, the negative exponent makes the answer less than one. Example: What is 5ˆ-2? 5ˆ-2 → 1/5² → 1/25 • Scientific notation rewrites very large or very small numbers using powers of 10. Example: 3,254,000 in scientific notation is 3.254 x 10ˆ6 .000000978 in scientific notation is 9.78 x 10ˆ-7 • If the number is smaller than 1, the exponent will be negative. If the number is larger than 1, the exponent will be positive. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
• Simplifying fractions uses the greatest common factor, or GCF. If a fraction is very large, look for the GCF of both the numerator and denominator. Then divide both the numerator and denominator by that factor. • When comparing and ordering fractions, the GCF is used to change the denominators of unlike fractions to denominators of like fractions. Example: Compare 6/16, 4/32 and 7/8 → 12/32, 4/32 and 28/32 • The fractions all can be changed to a denominator of 32. The fraction 6/16 becomes 12/32 and the fraction 7/8 becomes 28/32. When they have the same denominator, they can be compared, the smallest is 4/32, then 6/16 and the largest is 7/8. • A mixed number can be changed into a fraction by multiplying the denominator by the whole number and then adding the numerator, this number becomes the new numerator and the denominator stays the same. • To change an improper fraction into a mixed decimal, the numerator is divided by the denominator to get the whole number and the remainder. With the whole number, the remainder becomes the numerator and the denominator stays the same. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Try This! What is the prime factorization of 72, 96, and 384? Simplify the following fractions: 18/24 49/84 52/208 Order the following fractions from least to greatest: 1/2, 3/10, 14/15, 4/5 What is 8 2/3 as an improper fraction? What is 123/11 as a mixed number? Evaluate the following: 8³ 7º 5¹ 3ˆ-3 Evaluate the following: 2.36 x 10ˆ8 5.06 x 10ˆ-7 © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.