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Example 1 Use Divisibility Rules Example 2 Use Divisibility Rules to Solve a Problem Example 3 Find Factors of a Number Example 4 Identify Monomials Determine whether 435 is divisible by 2, 3, 5, 6, or 10. Number Divisible? Reason 2 no 3 yes 5 6 yes The ones digit is 5. no 10 no 435 is not divisible by 2, so it cannot be divisible by 6. The ones digit is not 0. The ones digit is 5 and 5 is not divisible by 2. The sum of the digits is or 12 and 12 is divisible by 3. Answer: So, 435 is divisible by 3 and 5. Determine whether 786 is divisible by 2, 3, 5, 6, or 10. Answer: 786 is divisible by 2, 3, and 6. Student Elections Sonya is running for student council president. She wants to give out campaign flyers with a pen to each student in the school. She can buy “Vote for Sonya” pens in packages of 5, 6, or 10. If there are 306 students in the school and she wants no pens left over, which size packages should she buy? Size Yes/No Reason 5 6 no The ones digit of 306 is not 0 or 5. yes 306 is divisible by 2 and 3, so it is also divisible by 6. Therefore, there would be no pens left over. The ones digit is not 0. no 10 Answer: Sonya should buy pens in packages of 6. Transportation A class of 72 students is taking a field trip. The transportation department can provide vans that seat 5, 6, or 10 students. If the teacher wants all vans to be the same size and no empty seats, what size vans should be used? Answer: Vans that seat 6 should be used. List all the factors of 64. Use the divisibility rules to determine whether 64 is divisible by 2, 3, 5, and so on. Then use division to find other factors of 64. Number 64 Divisible by Number? 2 yes yes 3 no 4 yes 5 no 6 no 7 no yes 1 8 Factor Pairs ___ ___ ___ ___ Answer: So, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64. List all the factors of 96. Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96 Determine whether is a monomial. Distributive Property Simplify. Answer: This expression is not a monomial because in its simplest form, it involves two terms that are added. Determine whether is a monomial. Answer: This expression is a monomial because it is the product of a rational number, and a variable, x. , Determine whether each expression is a monomial. a. Answer: monomial b. Answer: not a monomial Example 1 Write Expressions Using Exponents Example 2 Use Exponents in Expanded Form Example 3 Evaluate Expressions Write using exponents. Answer: The base is 6. It is a factor 4 times, so the exponent is 4. Write p using exponents. Answer: The base is p. It is a factor 1 time, so the exponent is 1. Write (–1)(–1)(–1) using exponents. Answer: The base is – 1. It is a factor 3 times, so the exponent is 3. Write using exponents. Answer: The base is exponent is 2. . It is a factor 2 times, so the Write each expression using exponents. Answer: First group the factors with like bases. Then write using exponents. Write each expression using exponents. d. a. Answer: Answer: b. e. Answer: Answer: c. Answer: Express 235,016 in expanded form. Answer: Step 1 Use place value to write the value of each digit in the number. Step 2 Write each place value as a power of 10 using exponents. Express 24,706 in expanded form. Answer: Evaluate . 4 is a factor two times. Multiply. Answer: 16 Evaluate if . Replace r with –2. –2 is a factor 3 times. Multiply. Subtract. Answer: Evaluate if and . Replace x with 2 and y with –2. Simplify the expression inside the parentheses. Evaluate (0)2. Simplify. Answer: 0 Evaluate each expression. a. Answer: 81 b. if Answer: 84 c. Answer: –24 Example 1 Simplify Fractions Example 2 Simplify Fractions Example 3 Simplify Fractions in Measurement Example 4 Simplify Algebraic Fractions Example 5 Simplify Algebraic Fractions Write in simplest form. Factor the numerator. Factor the denominator. The GCF of 16 and 24 is or 8. Divide the numerator and denominator by the GCF. Simplest form Answer: Write Answer: in simplest form. Write in simplest form. 1 1 1 1 1 1 1 1 Divide the numerator and the denominator by the GCF, . Simplify. Answer: Write Answer: in simplest form. Measurement 250 pounds is what part of 1 ton? There are 2000 pounds in 1 ton. Write the fraction 1 1 in simplest form. 1 1 1 1 1 1 Divide the numerator and the denominator by the GCF, . Simplify. Answer: So, 250 pounds is of a ton. 80 feet is what part of 40 yards? Answer: Simplify . 1 1 1 1 1 1 Divide the numerator and the denominator by the GCF, . Simplify. Answer: Simplify Answer: . Multiple-Choice Test Item Which fraction is A B written in simplest form? C D Read the Test Item In simplest form means that the GCF of the numerator and denominator is 1. Solve the Test Item 1 1 1 1 Factor. 1 1 Answer: C 1 1 Multiple-Choice Test Item Which fraction is written in simplest form? A C Answer: D B D Example 1 Multiply Powers Example 2 Multiply Monomials Example 3 Divide Powers Example 4 Divide Powers to Solve a Problem Find . The common base is 3. Add the exponents. Check Answer: Find Answer: . Find . The common base is y. Add the exponents. Answer: Find (3p4)(–2p3). (3 • –2)(p4 • p3) Use the Commutative and (3p4)(–2p3) Associative Properties. (–6)(p4+3) The common base is p. –6p7 Add the exponents. Answer: –6p7 Find each product. a. Answer: b. Answer: Find . The common base is 8. Subtract the exponents. Answer: Find . The common base is x. Subtract the exponents. Answer: Find Answer: Folding Paper If you fold a sheet of paper in half, you have a thickness of 2 sheets. Folding again, you have a thickness of 4 sheets. Continue folding in half and recording the thickness. How many times thicker is a sheet that has been folded 4 times than a sheet that has not been folded? Write a division expression to compare the thickness. Subtract the exponents. Answer: So, the paper is 16 times thicker. Racing Car A can run at a speed of miles per hour and car B runs at a speed of miles per hour. How many times faster is car A than car B? Answer: Car A is 2 times faster than car B. Example 1 Use Positive Exponents Example 2 Use Negative Exponents Example 3 Use Exponents to Solve a Problem Example 4 Algebraic Expressions with Negative Exponents Write using a positive exponent. Definition of negative exponent Answer: Write using a positive exponent. Definition of negative exponent Answer: Write each expression using a positive exponent. a. Answer: b. Answer: Write as an expression using a negative exponent. Find the prime factorization of 125. Definition of exponents Definition of negative exponent Answer: Write Answer: as an expression using a negative exponent. Physics An atom is an incredibly small unit of matter. The smallest atom has a diameter of approximately of a nanometer, or 0.0000000001 meter. Write the decimal as a fraction and as a power of 10. Write the decimal as a fraction. Definition of negative exponent Answer: Write 0.000001 as a fraction and as a power of 10. Answer: Evaluate . if Replace r with –4. Definition of negative exponent Find Answer: . Evaluate Answer: if . Example 1 Express Numbers in Standard Form Example 2 Express Numbers in Scientific Notation Example 3 Use Scientific Notation to Solve a Problem Example 4 Compare Numbers in Scientific Notation Express in standard form. Move the decimal point 4 places to the right. Answer: 43,950 Express in standard form. Move the decimal point 6 places to the left. Answer: 0.00000679 Express each number in standard form. a. Answer: 2,614,000 b. Answer: 0.000803 Express 800,000 in scientific notation. The decimal point moves 5 places. The exponent is positive. Answer: Express 1,320,000 in scientific notation. The decimal point moves 6 places. The exponent is positive. Answer: Express 0.0119 in scientific notation. The decimal point moves 2 places. The exponent is negative. Answer: Express each number in scientific notation. a. 65,000 Answer: b. 3,024,000 Answer: c. 0.00042 Answer: Space The table shows the planets and their distances from the Sun. Estimate how many times farther Pluto is from the Sun than Mercury is from the Sun. Planet Distance from the Sun (km) Mercury 5.80 x 107 Venus 1.03 x 108 Earth 1.55 x 108 Mars 2.28 x 108 Jupiter 7.78 x 108 Saturn 1.43 x 109 Uranus 2.87 x 109 Neptune 4.50 x 109 Pluto 5.90 x 109 Explore You know that the distance from the Sun to Pluto is km and the distance from the Sun to Mercury is km. Plan To find how many times farther Pluto is from the Sun than Mercury is from the Sun, find the ratio of Pluto’s distance to Mercury’s distance. Since you are estimating, round the distance to and round the distance to . Solve Divide Answer: So, Pluto is about 1.0 102 or 100 times farther from the Sun than Mercury. Examine Use estimation to check the reasonableness of the results. Space Use the table to estimate how many times farther Pluto is from the Sun than Earth is from the Sun. Planet Distance from the Sun (km) Mercury 5.80 x 107 Venus 1.03 x 108 Earth 1.55 x 108 Mars 2.28 x 108 Jupiter 7.78 x 108 Saturn 1.43 x 109 Uranus 2.87 x 109 Neptune 4.50 x 109 Pluto 5.90 x 109 Answer: 30 times farther Space The diameters of Mercury, Saturn, and Pluto are kilometers, kilometers, and kilometers, respectively. List the planets in order of increasing diameter. First, order the numbers according to their exponents. Then, order the numbers with the same exponent by comparing the factors. Mercury and Pluto Saturn Step 1 Compare the factors: Step 2 Pluto Mercury Answer: So, the order is Pluto, Mercury, and Saturn. Order the numbers , and in decreasing order. Answer: , , , , , and .