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Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Find each product. 1. 4 • 4 2. 7 • 7 3. 5 • 5 Perform the indicated operations. 5. 3 + 12 – 7 6. 6 • 1 ÷ 2 7. 4 – 2 + 9 8. 10 – 5 – 4 9. 5 • 5 + 7 10. 30 ÷ 6 • 2 1-2 4. 9 • 9 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Solutions 1. 4 • 4 = 16 2. 7 • 7 = 49 3. 5 • 5 = 25 4. 9 • 9 = 81 5. 3 + 12 – 7 = (3 + 12) – 7 = 15 – 7 = 8 6. 6 • 1 ÷ 2 = (6 • 1) ÷ 2 = 6 ÷ 2 = 3 7. 4 – 2 + 9 = (4 – 2) + 9 = 2 + 9 = 11 8. 10 – 5 – 4 = (10 – 5) – 4 = 5 – 4 = 1 9. 5 • 5 + 7 = (5 • 5) + 7 = 25 + 7 = 32 10. 30 ÷ 6 • 2 = (30 ÷ 6) • 2 = 5 • 2 = 10 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 32 + 62 – 14 • 3. 32 + 62 – 14 • 3 = 32 + 36 – 14 • 3 Simplify the power: 62 = 6 • 6 = 36. = 32 + 36 – 42 Multiply 14 and 3. = 68 – 42 Add and subtract in order from left to right. = 26 Subtract. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Evaluate 5x = 32 ÷ p for x = 2 and p = 3. 5x + 32 ÷ p = 5 • 2 + 32 ÷ 3 Substitute 2 for x and 3 for p. =5•2+9÷3 Simplify the power. = 10 + 3 Multiply and divide from left to right. = 13 Add. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Find the total cost of a pair of jeans that cost $32 and have an 8% sales tax. total cost original price C = p + sales tax r•p sales tax rate C=p+r•p = 32 + 0.08 • 32 Substitute 32 for p. Change 8% to 0.08 and substitute 0.08 for r. = 32 + 2.56 Multiply first. = 34.56 Then add. The total cost of the jeans is $34.56. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 3(8 + 6) ÷ (42 – 10). 3(8 + 6) ÷ (42 – 10) = 3(8 + 6) ÷ (16 – 10) Simplify the power. = 3(14) ÷ 6 Simplify within parentheses. = 42 ÷ 6 Multiply and divide from left to right. =7 Divide. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Evaluate each expression for x = 11 and z = 16. a. (xz)2 b. xz2 (xz)2 = (11 • 16)2 Substitute 11 for x and 16 for z. = (176)2 Simplify within parentheses. Multiply. = 30,976 Simplify. 1-2 xz2 = 11 • 162 = 11 • 256 = 2816 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify 4[(2 • 9) + (15 ÷ 3)2]. 4[(2 • 9) + (15 ÷ 3)2] = 4[18 + (5)2] First simplify (2 • 9) and (15 ÷ 3). = 4[18 + 25] Simplify the power. = 4[43] Add within brackets. = 172 Multiply. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 A carpenter wants to build three decks in the shape of regular hexagons. The perimeter p of each deck will be 60 ft. The perpendicular distance a from the center of each deck to one of the sides will be 8.7 ft. pa Use the formula A = 3 ( 2 pa A=3( 2 =3( ) to find the total area of all three decks. ) 60 • 8.7 ) 2 Substitute 60 for p and 8.7 for a. 522 Simplify the numerator. =3( 2 = 3(261) ) Simplify the fraction. = 783 Multiply. The total area of all three decks is 783 ft2. 1-2 Exponents and Order of Operations ALGEBRA 1 LESSON 1-2 Simplify each expression. 1. 50 – 4 • 3 + 6 44 2. 3(6 + 22) – 5 25 3. 2[(1 + 5)2 – (18 ÷ 3)] 60 Evaluate each expression. 4. 4x + 3y for x = 2 and y = 4 20 5. 2 • p2 + 3s for p = 3 and s = 11 51 6. xy2 + z for x = 3, y = 6 and z = 4 112 1-2 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 (For help, go to skills handbook page 725.) Write each decimal as a fraction and each fraction as a decimal. 1. 0.5 5. 2 5 2. 0.05 6. 3 8 3. 3.25 7. 1-3 2 3 4. 0.325 8. 3 5 9 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Solutions 1. 0.5 = 5 = 5 • 1 = 1 10 5•2 2 2. 0.05 = 5 = 5 • 1 = 1 100 5 • 20 20 3. 3.25 = 3 25 = 3 25 • 1 = 3 1 or 13 25 • 4 4 100 4 4. 0.325 = 325 = 25 • 13 = 13 1000 5. 25 • 40 40 2 = 2 ÷ 5 = 0.4 5 3 = 3 ÷ 8 = 0.375 8 7. 2 = 2 ÷ 3 = 0.6 3 8. 3 5 = 3 + (5 ÷ 9) = 3.5 9 6. 1-3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Name the set(s) of numbers to which each number belongs. a. –13 b. 3.28 integers rational numbers rational numbers 1-3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Which set of numbers is most reasonable for displaying outdoor temperatures? integers 1-3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Determine whether the statement is true or false. If it is false, give a counterexample. All negative numbers are integers. A negative number can be a fraction, such as – The statement is false. 1-3 2 . This is not an integer. 3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Write – 3 , – 7 , and – 5 , in order from least to greatest. 4 12 8 – 3 = –0.75 Write each fraction as a decimal. –0.75 < –0.625 < –0.583 Order the decimals from least to greatest. 4 – 7 = –0.583 12 – 5 = –0.625 8 From least to greatest, the fractions are – 3 , – 5 , and – 7 . 4 1-3 8 12 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Find each absolute value. a. |–2.5| b. |7| –2.5 is 2.5 units from 0 on a number line. 7 is 7 units from 0 on a number line. |–2.5| = 2.5 |7| = 7 1-3 Exploring Real Numbers ALGEBRA 1 LESSON 1-3 Name the set(s) of numbers to which each given number belongs. 1. –2.7 2. rational numbers 11 3. 16 irrational numbers Use <, =, or > to compare. 4. 3 4 > 5. 5 8 –3 < – 5 4 6. Find |– 7 |. 12 7 12 1-3 8 natural numbers, whole numbers integers, rational numbers Multiplying and Dividing Real Numbers ALGEBRA 1 LESSON 1-6 x Evaluate – y – 4z2 for x = 4, y = –2, and z = –4. –4 – x – 4z2 = –2 – 4(–4)2 y –4 Substitute 4 for x, –2 for y, and –4 for z. = –2 – 4(16) Simplify the power. = 2 – 64 Divide and multiply. = –62 Subtract. 1-6 Multiplying and Dividing Real Numbers ALGEBRA 1 LESSON 1-6 Evaluate p for p = 3 and r = – 3 . r p =p÷r r 2 4 Rewrite the equation. 3 (– 34 ) Substitute 2 for p and – 4 for r. 3 4 Multiply by – 3 , the reciprocal of – 4 . = 2 ÷ = 2 (– 3 = –2 ) 3 3 4 Simplify. 1-6 3 Multiplying and Dividing Real Numbers ALGEBRA 1 LESSON 1-6 Simplify. 1. –8(–7) 2. –6(–7 + 10) – 4 – 22 56 Evaluate each expression for m = –3, n = 4, and p = –1. 3. 8m + p n –7 4. (mp)3 5. mnp 27 12 1 2 6. Evaluate 2a ÷ 4b – c for a = –2, b = – 1 , and c = – . 3 1 32 1-6 The Distributive Property ALGEBRA 1 LESSON 1-7 (For help, go to Lessons 1-2 and 1-6.) Use the order of operations to simplify each expression. 1. 3(4 + 7) 2. –2(5 + 6) 4. –0.5(8 – 6) 5. 1 t(10 – 4) 2 1-7 3. –1(–9 + 8) 6. m(–3 – 1) The Distributive Property ALGEBRA 1 LESSON 1-7 Solutions 1. 3(4 + 7) = 3(11) = 33 2. –2(5 + 6) = –2(11) = –22 3. –1(–9 + 8) = –1(–1) = 1 4. –0.5(8 – 6) = –0.5(2) = –1 5. 1 1 1 1 t(10 – 4) = t(6) = (6)t = ( • 6) t = 3t 2 2 2 2 6. m(–3 – 1) = m(–4) = –4m 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Use the Distributive Property to simplify 26(98). 26(98) = 26(100 – 2) Rewrite 98 as 100 – 2. = 26(100) – 26(2) Use the Distributive Property. = 2600 – 52 Simplify. = 2548 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Find the total cost of 4 CDs that cost $12.99 each. 4(12.99) = 4(13 – 0.01) Rewrite 12.99 as 13 – 0.01. = 4(13) – 4(0.01) Use the Distributive Property. = 52 – 0.04 Simplify. = 51.96 The total cost of 4 CDs is $51.96. 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Simplify 3(4m – 7). 3(4m – 7) = 3(4m) – 3(7) = 12m – 21 Use the Distributive Property. Simplify. 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Simplify –(5q – 6). –(5q – 6) = –1(5q – 6) Rewrite the expression using –1. = –1(5q) – 1(–6) Use the Distributive Property. = –5q + 6 Simplify. 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Simplify –2w2 + w2. –2w2 + w2 = (–2 + 1)w2 = –w2 Use the Distributive Property. Simplify. 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Write an expression for the product of –6 and the quantity 7 minus m. Relate: –6 times the quantity 7 minus m Write: –6 • (7 – m) –6(7 – m) 1-7 The Distributive Property ALGEBRA 1 LESSON 1-7 Simplify each expression. 3. – 3(2y – 7) – 6y + 21 1. 11(299) 3289 2. 4(x + 8) 4x + 32 4. –(6 + p) 5. 1.3a + 2b – 4c + 3.1b – 4a –6–p –2.7a + 5.1b – 4c 6. Write an expression for the product of 4 and the quantity b minus 3 . 7 4 3 b – ( ) 7 5 1-7 5 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 (For help, go to Lessons 1-4 and 1-6.) Simplify each expression. 1. 8 + (9 + 2) 2. 3 • (–2 • 5) 3. 7 + 16 + 3 4. –4(7)(–5) 5. –6 + 9 + (–4) 6. 0.25 • 3 • 4 7. 3 + x – 2 8. 2t – 8 + 3t 9. –5m + 2m – 4m 1-8 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 Solutions 1. 8 + (9 + 2) = 8 + (2 + 9) = (8 + 2) + 9 = 10 + 9 = 19 2. 3 • (–2 • 5) = 3 • (–10) = –30 3. 7 + 16 + 3 = 7 + 3 + 16 = 10 + 16 = 26 4. –4(7)(–5) = –4(–5)(7) = 20(7) = 140 5. –6 + 9 + (–4) = –6 + (–4) + 9 = –10 + 9 = –1 6. 0.25 • 3 • 4 = 0.25 • 4 • 3 = 1 • 3 = 3 7. 3 + x – 2 = 3 + (–2) + x = 1 + x 8. 2t – 8 + 3t = 2t + 3t – 8 = (2 + 3)t – 8 = 5t – 8 9. –5m + 2m – 4m = (–5 + 2 – 4)m = –7m 1-8 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 Name the property each equation illustrates. a. 3 • a = a • 3 Commutative Property of Multiplication, because the order of the factors changes b. p • 0 = 0 Multiplication Property of Zero, because a factor multiplied by zero is zero c. 6 + (–6) = 0 Inverse Property of Addition, because the sum of a number and its inverse is zero 1-8 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 Suppose you buy a shirt for $14.85, a pair of pants for $21.95, and a pair of shoes for $25.15. Find the total amount you spent. 14.85 + 21.95 + 25.15 = 14.85 + 25.15 + 21.95 Commutative Property of Addition = (14.85 + 25.15) + 21.95 Associative Property of Addition = 40.00 + 21.95 Add within parentheses first. = 61.95 Simplify. The total amount spent was $61.95. 1-8 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 Simplify 3x – 4(x – 8). Justify each step. 3x – 4(x – 8) = 3x – 4x + 32 Distributive Property = (3 – 4)x + 32 Distributive Property = –1x + 32 Subtraction = –x + 32 Identity Property of Multiplication 1-8 Properties of Real Numbers ALGEBRA 1 LESSON 1-8 Name the property that each equation illustrates. 1. 1m = m 2. (– 3 + 4) + 5 = – 3 + (4 + 5) Iden. Prop. Of Mult. 3. –14 • 0 = 0 Assoc. Prop. Of Add. Mult. Prop. Of Zero 4. Give a reason to justify each step. a. 3x – 2(x + 5) = 3x – 2x – 10 Distributive Property b. = 3x + (– 2x) + (– 10) Definition of Subtraction c. = [3 + (– 2)]x + (– 10) Distributive Property d. = 1x + (– 10) Addition e. = 1x – 10 Definition of Subtraction f. = x – 10 Identity Property of Multiplication 1-8