Logarithms Exponential Equations: Logarithmic Equations: Exponent Base Exponent What it equals.
Lesson 2 Reteach - Miss Jarrelljarrellmnps.weebly.com/.../recoverych1.l2-5reteach.pdfLesson 5...
Transcript of Lesson 2 Reteach - Miss Jarrelljarrellmnps.weebly.com/.../recoverych1.l2-5reteach.pdfLesson 5...
Course 3 • Chapter 1 Real Numbers 3
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Lesson 2 ReteachPowers and Exponents
Example 1Write 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 Commutative Property
= (y ! y ! y) · (x · x) Associative Property
= y3 ! x2 Defi nition of exponents
To evaluate a power, perform the repeated multiplication to fi nd the product.
Example 2
Evaluate (-6)4.
(- 6)4 = (- 6) ! (- 6) ! (- 6) ! (- 6) Write the power as a product.
= 1,296 Multiply.
The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction.
Example 3Evaluate m2 + (n - m)3 if m = -3 and n = 2.
m2 + (n - m)3 = (- 3)2 + (2 - (- 3))3 Replace m with - 3 and n with 2.
= (- 3)2 + (5)3 Perform operations inside parentheses.
= (- 3 ! - 3) + (5 ! 5 ! 5) Write the powers as products.
= 9 + 125 or 134 Add.
ExercisesWrite each expression using exponents.
1. 8 ! 8 ! 8 ! 8 ! 8 2. a ! a ! a ! a ! a ! a 3. 5 ! 5 ! 9 ! 9 ! 5 ! 9 ! 5 ! 5
Evaluate each expression.
4. 24 5. (- 3)5 6. ( 3 − 4 )
3
ALGEBRA Evaluate each expression if a = 5 and b = - 4.
7. a2 + b2 8. (a + b)2 9. a + b2
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.
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Course 3 • Chapter 1 Real Numbers 5
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Lesson 3 ReteachMultiply and Divide Monomials
The Product of Powers rule states that to multiply powers with the same base, add their exponents.
Example 1Simplify. Express using exponents.
a. 23 ! 22
23 ! 22 = 23 + 2 The common base is 2.
= 25 Add the exponents.
b. 2s6(7s7)
2s6(7s7) = (2 ! 7)(s6 ! s7) Commutative and Associative Properties
= 14(s6 + 7) The common base is s.
= 14s13 Add the exponents.
The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.
Example 2Simplify k
8 −−
k . Express using exponents.
k8 −−
k1 = k8 - 1 The common base is k.
= k7 Subtract the exponents.
Example 3
Simplify (- 2)10 ! 56 ! 63
−−−−−−−− (- 2)6 ! 53 ! 62
.
(- 2)10 ! 56 ! 63
−−−−−−−−− (- 2)6 ! 53 ! 62 = ( (- 2)10
−−−− (- 2)6 ) ! ( 5
6 −−
53 ) ! ( 63 −−
62 ) Group by common base.
= (–2)4 ! 53 ! 61 Subtract the exponents.
= 16 ! 125 ! 6 or 12,000 Simplify.
ExercisesSimplify. Express using exponents.
1. 52 ! 55 2. e2 ! e7 3. 2a5 ! 6a 4. 4x2(–5x6)
5. 79 −−
73 6. v14 −−
v6 7. 15w7 −−−−
5w2 8. 10m8 −−−−
2m
9. 25 ! 37 ! 43
−−−−−− 21 ! 35 ! 4
10. 415 ! (- 5)6
−−−−−− 412 ! (- 5)4 11. 6
7 ! 76 ! 85 −−−−−−
65 ! 75 ! 84 12. (- 3)6 ! 105
−−−−−−− (- 3)4 ! 103
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Course 3 • Chapter 1 Real Numbers 7
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Lesson 4 ReteachPowers of Monomials
Example 1 Simplify (53)6.
(53)6 = 53 · 6 Power of a power
= 518 Simplify.
Example 2
Simplify (-3m2n4)3.
(-3m2n4)3 = (-3)3 · m2 · 3 · n4 · 3 Power of a product
= -27m6n12 Simplify.
ExercisesSimplify.
1. (43)5 2. (42)7 3. (92)4
4. (k4)2 5. [(63)2]2 6. [(32)2]3
7. (5q4r2)5 8. (3y2z2)6 9. (7a4b3c7)2
10. (-4d3e5)2 11. (-5g4h9)7 12. (0.2k8)2
Power of a Power: To fi nd the power of a power, multiply the exponents.
Power of a Product: To fi nd the power of a product, fi nd the power of each factor and multiply.
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Course 3 • Chapter 1 Real Numbers 11
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Lesson 5 ReteachNegative Exponents
Example 1Write each expression using a positive exponent.
a. 7−3
7−3 = 1 −− 73 Defi nition of negative exponent
b. a−4
a−4 = 1 −− a4 Defi nition of negative exponent
Example 2Evaluate each expression.
a. 5−4
5−4 = 1 −− 54 Defi nition of negative exponent
= 1 −−− 625
54 = 5 · 5 · 5 · 5
b. (−3)−5
(−3)−5 = 1 −−−− (−3)5 Defi nition of negative exponent
= 1 −−−− −243
(−3)5 = (−3) · (−3) · (−3) · (−3) · (−3)
Example 3Write 1 −−
65 as an expression using a negative exponent.
1 −− 65 = 6−5 Defi nition of negative exponent
Example 4Simplify. Express using positive exponents.
a. x−3 · x5
x−3· x5 = x(−3) + 5 Product of Powers
= x2 Add the exponents.
b. w−5 −−−
w−7
w−5 −−−
w−7 = w−5 − (−7) Quotient of Powers
= w2 Subtract the exponents.
ExercisesWrite each expression using a positive exponent.
1. a−8 2. 6−3 3. n−4
Evaluate each expression.
4. 7−2 5. 9−3 6. (−2)−5
Write each fraction as an expression using a negative exponent.
7. 1 −− 57 8. 1 −−
36 9. 1 −− x8
Simplify. Express using positive exponents.
10. 4−2 · 4−4 11. r−3 · r5 12. h−2 −−−
h4
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.
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