Lesson 1 Contents Example 1Use Divisibility Rules Example 2Use Divisibility Rules to Solve a Problem Example 3Find Factors of a Number Example 4Identify Monomials

Example 1-1a Determine whether 435 is divisible by 2, 3, 5, 6, or 10. NumberDivisible? Reason no The ones digit is 5 and 5 is not divisible by 2. The ones digit is 5. yesno The ones digit is not 0. no 435 is not divisible by 2, so it cannot be divisible by 6. Answer: So, 435 is divisible by 3 and 5. yes The sum of the digits is or 12 and 12 is divisible by 3.

Example 1-1b Determine whether 786 is divisible by 2, 3, 5, 6, or 10. Answer: 786 is divisible by 2, 3, and 6.

Example 1-2a 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? SizeYes/No Reason 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 Answer: Sonya should buy pens in packages of 6.

Example 1-2b 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.

Example 1-3a 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. Number64 Divisible by Number?Factor Pairs ___ no ___ no ___ no ___ no yes

Example 1-3b Answer: So, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

Example 1-3c 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. Example 1-4a Simplify. Answer: This expression is not a monomial because in its simplest form, it involves two terms that are added. Distributive Property

Example 1-4b Determine whether is a monomial. Answer:This expression is a monomial because it is the product of a rational number,, and a variable, x.

Example 1-4c Determine whether each expression is a monomial. Answer: monomial Answer: not a monomial a. b.

End of Lesson 1

Lesson 2 Contents Example 1Write Expressions Using Exponents Example 2Use Exponents in Expanded Form Example 3Evaluate Expressions

Write using exponents. Example 2-1a Answer:The base is 6. It is a factor 4 times, so the exponent is 4.

Example 2-1b Write p using exponents. Answer:The base is p. It is a factor 1 time, so the exponent is 1.

Example 2-1c 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. Example 2-1d Answer:The base is. It is a factor 2 times, so the exponent is 2.

Example 2-1e Write each expression using exponents. Answer:First group the factors with like bases. Then write using exponents.

Example 2-1f Answer: Write each expression using exponents. a. b. c. d. e.

Example 2-2a Express 235,016 in expanded form. Answer: Step 1Use 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.

Example 2-2b Express 24,706 in expanded form. Answer:

Example 2-3a Answer: 16 4 is a factor two times. Multiply. Evaluate.

Example 2-3b –2 is a factor 3 times. Multiply. Subtract. Answer: Replace r with –2. Evaluate.if

Example 2-3c Simplify the expression inside the parentheses. Evaluate (0) 2. Replace x with 2 and y with –2. Simplify. Answer: 0 Evaluate. ifand

Example 2-3d Evaluate each expression. Answer: 81 Answer: 84 Answer: –24 a. b. if c.

End of Lesson 2

Lesson 5 Contents Example 1Simplify Fractions Example 2Simplify Fractions Example 3Simplify Fractions in Measurement Example 4Simplify Algebraic Fractions Example 5Simplify Algebraic Fractions

Example 5-1a Write in simplest form. The GCF of 16 and 24 is or 8. Answer: Factor the numerator. Factor the denominator. Divide the numerator and denominator by the GCF. Simplest form

Example 5-1b Write in simplest form. Answer:

Example 5-2a Write in simplest form. Simplify. Answer: Divide the numerator and the denominator by the GCF,

Example 5-2b Write in simplest form. Answer:

Example 5-3a Measurement 250 pounds is what part of 1 ton? There are 2000 pounds in 1 ton. Write the fraction in simplest form. Answer: So, 250 pounds is of a ton. Simplify. Divide the numerator and the denominator by the GCF,

Example 5-3b 80 feet is what part of 40 yards? Answer:

Example 5-4a Simplify. Answer: 11 Divide the numerator and the denominator by the GCF,

Example 5-4b Simplify. Answer:

Example 5-5a ABCDABCD Read the Test Item In simplest form means that the GCF of the numerator and denominator is 1. Which fraction is written in simplest form? Multiple-Choice Test Item

Example 5-5b Solve the Test Item Answer: C Factor

Example 5-5c ABCDABCD Answer: D Which fraction is written in simplest form? Multiple-Choice Test Item

End of Lesson 5

Lesson 6 Contents Example 1Multiply Powers Example 2Multiply Monomials Example 3Divide Powers Example 4Divide Powers to Solve a Problem

Example 6-1a Find. Check Answer: Add the exponents. The common base is 3.

Example 6-1b Find. Answer:

Find. Example 6-2a Answer: The common base is y. Add the exponents.

Example 6-2b Find (3p 4 )(–2p 3 ). Answer: –6p 7 Use the Commutative and Associative Properties. (3p 4 )(–2p 3 )(3 –2)(p 4 p 3 ) Add the exponents. –6p 7 The common base is p. (–6)(p 4+3 )

Example 6-2c Find each product. Answer: a. b. Answer:

Example 6-3a The common base is 8. Answer: Subtract the exponents. Find.

Example 6-3b The common base is x. Subtract the exponents. Answer: Find.

Example 6-3c Find Answer:

Example 6-4a 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.

Example 6-4b Racing Car A can run at a speed ofmiles 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.

End of Lesson 6

Lesson 7 Contents Example 1Use Positive Exponents Example 2Use Negative Exponents Example 3Use Exponents to Solve a Problem Example 4Algebraic Expressions with Negative Exponents

Example 7-1a Answer: Definition of negative exponent Write using a positive exponent.

Example 7-1b Answer: Write using a positive exponent. Definition of negative exponent

Example 7-1c Write each expression using a positive exponent. Answer: a. b. Answer:

Example 7-2a Write as an expression using a negative exponent. Answer: Find the prime factorization of 125. Definition of exponents Definition of negative exponent

Example 7-2b Write as an expression using a negative exponent. Answer:

Example 7-3a Physics An atom is an incredibly small unit of matter. The smallest atom has a diameter of approximately of a nanometer, or meter. Write the decimal as a fraction and as a power of 10. Answer: Write the decimal as a fraction. Definition of negative exponent

Example 7-3b Write as a fraction and as a power of 10. Answer:

Example 7-4a Find. Answer: Replace r with –4. Definition of negative exponent Evaluate.if

Example 7-4b Answer: Evaluate. if

End of Lesson 7

Lesson 8 Contents Example 1Express Numbers in Standard Form Example 2Express Numbers in Scientific Notation Example 3Use Scientific Notation to Solve a Problem Example 4Compare Numbers in Scientific Notation

Express in standard form. Example 8-1a Answer: 43,950 Move the decimal point 4 places to the right.

Example 8-1b Answer: Move the decimal point 6 places to the left. Express in standard form.

Example 8-1c Express each number in standard form. a. b. Answer: 2,614,000 Answer:

Example 8-2a Express 800,000 in scientific notation. Answer: The exponent is positive. The decimal point moves 5 places.

Example 8-2b Express 1,320,000 in scientific notation. The exponent is positive. The decimal point moves 6 places. Answer:

Example 8-2c Express in scientific notation. The exponent is negative. The decimal point moves 2 places. Answer:

Example 8-2d Express each number in scientific notation. a. 65,000 b. 3,024,000 c Answer:

Example 8-3a 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. PlanetDistance from the Sun (km) Mercury 5.80 x 10 7 Venus 1.03 x 10 8 Earth1.55 x 10 8 Mars2.28 x 10 8 Jupiter7.78 x 10 8 Saturn1.43 x 10 9 Uranus2.87 x 10 9 Neptune4.50 x 10 9 Pluto5.90 x 10 9

Example 8-3a Explore You know that the distance from the Sun to Pluto iskm and the distance from the Sun to Mercury iskm. 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.

Example 8-3b Examine Use estimation to check the reasonableness of the results. Solve Divide Answer: So, Pluto is about 1.0  10 2 or 100 times farther from the Sun than Mercury.

PlanetDistance from the Sun (km) Mercury 5.80 x 10 7 Venus 1.03 x 10 8 Earth1.55 x 10 8 Mars2.28 x 10 8 Jupiter7.78 x 10 8 Saturn1.43 x 10 9 Uranus2.87 x 10 9 Neptune4.50 x 10 9 Pluto5.90 x 10 9 Example 8-3c Space Use the table to estimate how many times farther Pluto is from the Sun than Earth is from the Sun. Answer: 30 times farther

Example 8-4a Space The diameters of Mercury, Saturn, and Pluto arekilometers,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.

Example 8-4b Answer: So, the order is Pluto, Mercury, and Saturn. Step 1 Step 2 Mercury and Pluto Saturn Pluto Mercury Compare the factors:

Example 8-4c Order the numbers,,, andin decreasing order. Answer:,,, and.

End of Lesson 8