Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
NAME _____________________________________________ DATE ____________________________ PERIOD _____________ Lesson 1 Reteach Powers and Exponents The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent tells how many times the base is used as a factor. Example 1 Write each expression using exponents. a. 7 • 7 • 7 • 7 7 • 7 • 7 • 7 = 74 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent. b. y • y • x • y • x y•y•x•y•x=y•y•y•x•x = (y • y • y) • (x • x) = y3 • x2 Commutative Property Associative Property Definition of exponents To evaluate a power, perform the repeated multiplication to find the product. Example 2 Evaluate (–64). (–6)4 = (–6) • (–6) • (–6) • (–6) = 1,296 Write the power as a product. Multiply. The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction. Example 3 Evaluate m2 + (n – m)3 if m = –3 and n = 2. m2 + (n – m)3 = (–3)2 + (2 – (–3))3 2 = (–3) + (5) 3 Replace m with –3 and n with 2. Perform operations inside parentheses. = (–3 • –3) + (5 • 5 • 5) Write the powers as products. = 9 + 125 or 134 Add. Lesson 2 Reteach Multiply and Divide Monomials The Product of Powers rule states that to multiply powers with the same base, add their exponents. Example 1 Simplify. Express using exponents. a. 𝟐𝟑 • 𝟐𝟐 b. 𝟐𝒔𝟔 (𝟕𝒔𝟕 ) 23 • 22 = 23+2 The Common base 2. = 25 Add the exponents. 2𝑠 6 (7𝑠 7 ) = (2 • 7)(s 6 • s 7 ) = 14(𝑠 6+7 ) Commutative and Associative Properties The common base is s. 3 = 14𝑠 13 Add the exponents. NAME _____________________________________________ DATE ____________________________ PERIOD _____________ The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents. Example 2 Simplify 𝒌𝟖 . 𝒌 Express using exponents. 𝑘8 𝑘1 = 𝑘 8−1 The common base is k. = 𝑘7 Subtract the exponents. Example 3 Simplify (−𝟐)𝟏𝟎 • 𝟓𝟔 • 𝟔𝟖 (−𝟐)𝟔 • 𝟓𝟖 • 𝟔𝟐 (−2)10 • 56 • 68 (−2)6 • 58 • 62 . (−2)10 ) (−2)6 = ( 56 5 63 6 • ( 3) • ( 2) Group by common base. = (−2)4 • 53 • 61 Subtract the exponents. = 16 • 125 • 6 or 12,000 Simplify. Lesson 3 Reteach Powers of Monomials Power of a Power: To find the power of a power, multiply the exponents. Power of a Product: To find the power of a product, find the power of each factor and multiply. Example 1 Simplify (𝟓𝟑 )𝟔 . (53 )6 = 53 • 6 Power of a power = 518 Simplify. Example 2 Simplify (−𝟑𝒎𝟐 𝒏𝟒 )𝟖 . (−3𝑚2 𝑛4 )3 = (−3)3 • 𝑚2 • 3 • 𝑛4 • 3 6 12 = −27𝑚 𝑛 Power of a product Simplify. Lesson 4 Reteach Negative Exponents Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of the number to the nth power. Example 1 3 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ Write each expression using a positive exponent. a. 𝟕−𝟑 1 7−3 = 73 b. 𝒂−𝟒 1 𝑎−4 = 𝑎3 Definition of negative exponent Definition of negative exponent Example 2 Evaluate each expression. a. 𝟓−𝟒 1 5−4 = 54 = 1 625 b. (−𝟑)−𝟓 1 (−3)−5 = (−3)5 Definition of negative exponent = 54 = 5 • 5 • 5 • 5 1 −243 Definition of negative exponent (–3)5 = (−3) • (−3) • (−3) • (−3) • (−3) Example 3 𝟏 Write 𝟔𝟓 as an expression using a negative exponent. 1 65 = 6−5 Definition of negative exponent Example 4 Simplify. Express using positive exponents. 𝒘−𝟓 a. 𝒙−𝟑 • 𝒙𝟓 b. 𝒘−𝟕 𝑥 −3 • 𝑥 5 = 𝑥 (−3)+5 = 𝑥2 Product of Powers Add the exponents. 𝑤 −5 𝑤 −7 = 𝑤 −5−(−7) = 𝑤2 Quotient of Powers Subtract the exponents. Lesson 5 Reteach Scientific Notation A number in scientific notation is written as the product of a factor that is at least one but less than ten and a power of ten. Example 1 Write 𝟖. 𝟔𝟓 × 𝟏𝟎𝟕 in standard form. 8.65 × 107 = 8.65 × 10,000,000 = 86,500,000 107= 10 • 10 • 10 • 10 • 10 • 10 • 10 or 10,000,000 The decimal point moves 7 places to the right. Example 2 Write 𝟗. 𝟐 × 𝟏𝟎−𝟑 in standard form. 9.2 × 10−3 = 9.2 × 0.001 The decimal point moves 3 places to the left. = 0.0092 Example 3 Write 76,250 in scientific notation. 3 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 76,250 = 7.625 × 10,000 The decimal point moves 4 places. = 7.625 × 104 Since 76,250 is >1, the exponent is positive. Example 4 Write 0.00157 in scientific notation. 0.00157 = 1.57 × 0.001 The decimal point moves 3 places. = 1.57 × 10−3 Since 0.00157 is <1, the exponent is negative. Lesson 6 Reteach Compute with Scientific Notation You can use the Product of Powers and Quotient of Powers properties to multiply and divide numbers written in scientific notation. Example 1 Evaluate (𝟑. 𝟒 × 𝟏𝟎𝟓 )(𝟐. 𝟑 × 𝟏𝟎𝟑 ). Express the result in scientific notation. (3.4 × 105 )(2.3 × 103 ) = (3.4 × 2.3)(105 × 103 ) = (7.82)(105 × 103 ) = 7.82 × 10 5+3 Commutative and Associative Properties Multiply 3.4 by 2.3. Product of Powers = 7.82 × 108 Add the exponents. Example 2 Evaluate 2.325 × 104 3.1 × 104 𝟐.𝟑𝟐𝟓 × 𝟏𝟎𝟒 𝟑.𝟏 × 𝟏𝟎𝟒 . Express the result in scientific notation. 2.325 104 ) ( ) 3.1 102 =( 104 Associative Property = (0.75) (102 ) Divide 2.325 by 3.1. = 0.75 × 104−2 Quotient of Powers = 0.75 × 10 2 Subtract the exponents. = 0.75 × 102 Write 0.75 × 102 in scientific notation. = 7.5 × 10 Since the decimal point moved 1 place to the right, subtract 1 from the exponent. Example 3 Evaluate (𝟓. 𝟐𝟒 × 𝟏𝟎𝟓 ) + (𝟖. 𝟔𝟓 × 𝟏𝟎𝟔 ). Express the result in scientific notation. (5.24 × 105 ) + (8.65 × 106 ) = (5.24 × 105 ) + (86.5 × 105 ) = (5.24 + 86.5) × 105 = 91.74 × 10 5 = 9.174 × 106 Write 8.65 × 106 as 86.5 × 105. Distributive Property Add 5.24 and 86.5. Write 91.74 × 105 in scientific notation. 3