4-1 exponents repeated multiplication or division using bases and exponents
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4-1 Exponents
Repeated Multiplicationor Division
Using Bases and Exponents
4-1 Exponents
Vocabulary/Essential Question
EQ: How can we perform repeated multiplication and division in a shortened form?
exponential form
exponent
base
power
4-1 Exponents
If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power. Both 27 and 33 represent the same power.
7
ExponentBase
2
4-1 Exponents
.
• Simple exponent examples
4-3 Properties of Exponents
The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product.
7 • 7 • 7 • 7 = 74
(7 • 7 • 7) • 7 = 73 • 71 = 74
(7 • 7) • (7 • 7) = 72 • 72 = 74
4-1 Exponents
Identify how many times 4 is a factor.4 • 4 • 4 • 4 = 44
Write in exponential form.
Additional Example 1: Writing Exponents
A. 4 • 4 • 4 • 4
Read –(63) as “ negative 6 to the 3rd power” or “negative 6 cubed”.
Reading Math
Identify how many times –6 is a factor.
(–6) • (–6) • (–6) = (–6)3
B. (–6) • (–6) • (–6)
4-1 Exponents
Identify how many times 5 and d are used as a factor.
Additional Example 1: Writing Exponents
C. 5 • 5 • d • d • d • d
Write in exponential form.
5 • 5 • d • d • d • d = 52d4
4-1 Exponents.
• Bases with negative exponents or exponents with negative bases.
4-1 Exponents
Look for a pattern in the table to extend what you know about exponents to include negative exponents.
÷ 10 ÷ 10 ÷ 10 ÷ 10
102 101 100 10–1 10–2
10 • 10
100
10
10 1
1 110
110
= 0.1
110 • 10
1100
= 0.01
4-1 Exponents
D.
= 256
= (–4) • (–4) • (–4) • (–4) (–4)4
C. (–4)4
Simplify.
Additional Example 2: Simplifying Powers
Find the product of four –4’s.
Find the product of eight 1/2’s.
4-1 Exponents
D. 97
= –25
97 = 9 • 9 • 9 • 9 • 9 • 9 • 9
= 4,782,969
= –(5) • (5) –(5)2
C. –(5)2
Simplify.
Check It Out: Example 2
Find the product of two 5’s and then make the answer negative.
Find the product of seven 9’s.
4-1 Exponents
Additional Example 1: Using a Pattern to Simplify Negative Exponents
Simplify. Write in decimal form.
A. 10–2
10–2 = 1
10 • 10
= 1100
= 0.01
B. 10–1
= = 0.1110
= 110
Extend the pattern from the table.
Multiply. Write as a decimal.
Extend the pattern from the table.
Multiply. Write as a decimal.
4-1 Exponents
Check It Out: Example 1A
10–8
=1
100,000,000
= 1
10 • 10 • 10 • 10 • 10 • 10 • 10 • 10
Extend the pattern from the table.
= 0.00000001
Multiply.
Write as a decimal.
Simplify. Write in decimal form.
4-1 Exponents
5–3
Write the power under 1; change the sign of the exponent.
Additional Example 2A: Evaluating Negative Exponents
Simplify.
Find the product of three ’s.1 5
Simplify.
4-1 Exponents
(–10)–3 Write the power under 1; change the sign of the exponent.
Additional Example 2B: Evaluating Negative Exponents
Simplify.
Find the product of three ’s. 1 –10
Simplify.
1–10 • –10 • –10
–1000 1
= –0.001
4-1 Exponents.
• Multiplying powers with the same base.
4-3 Properties of Exponents
Additional Example 1: Multiplying Powers with the Same Base
A. 66 • 63
69
66 + 3
B. n5 • n7
n12
n5 + 7
Add exponents.
Add exponents.
Multiply. Write the product as one power.
4-3 Properties of Exponents
D. 244 • 244
C. 25 • 2
2 6
25 + 1
248
24 4 + 4
Think: 2 = 2 1
Additional Example 1: Multiplying Powers with the Same Base Continued
Multiply. Write the product as one power.
Add exponents.
Add exponents.
4-3 Properties of Exponents
Check It Out: Example 1
A. 42 • 44
46
42 + 4
B. x2 • x3
x5
x2 + 3
Add exponents.
Add exponents.
Multiply. Write the product as one power.
4-3 Properties of Exponents
Notice what occurs when you divide powers with the same base.
5 5 555
53=
5 5 5 5 5= 5 5 = 52=
5 5 55 5 5 5 5
4-3 Properties of Exponents
.
• Dividing powers with the same base.
4-3 Properties of Exponents
Subtract exponents.
72
75 – 3
75
73
Additional Example 2: Dividing Powers with the Same Base
Divide. Write the quotient as one power.
A.
x10
x9B.
Subtract exponents.x10 – 9
x Think: x = x1
4-3 Properties of Exponents
Subtract exponents.
97
99 – 2
99
92
Check It Out: Example 2
Divide. Write the product as one power.
A.
B. e10
e5
Subtract exponents.e10 – 5
e5
4-3 Properties of Exponents
.
• Raising a power to a power
4-3 Properties of Exponents
Simplify.
Multiply exponents.
Additional Example 3: Raising a Power to a Power
A. (54)2
(54)2
54 • 2
58
B. (67)9
(67)9
67 • 9
663
Multiply exponents.
4-3 Properties of Exponents
Simplify.
Multiply exponents.
Additional Example 3: Raising a Power to a Power
C. D. (172)–20
(172)–20
172 • –20
17–40
Multiply exponents.
2 3
12 –3
2 3
12 • –3
4-3 Properties of Exponents
Simplify.
Multiply exponents.
Check It Out: Example 3
A. (33)4
(33)4
33 • 4
312
B. (48)2
(48)2
48 • 2
416
Multiply exponents.
4-3 Properties of Exponents
Simplify.
Multiply exponents.
Check It Out: Example 3
C. D. (134)–10
(134)–10
134 • –10
13–40
Multiply exponents.
1 4
11 –2
1 4
11• –2
4-3 Properties of Exponents
Lesson Quiz
Write the product or quotient as one power.
3.
8 9n 71. n3 n4
109
105 10 4 4.
t 2
5. 32 • 33 • 35 3 10
2. 8 • 88
t9
t7
6. (m2)19 m38
7. (9-8)9 9–72 8. (104)0 1
4-3 Properties of Exponents
2. Write the product as one power.
44 43 42
A. 49
B. 414
C. 424
D. 4432
Lesson Quiz for Student Response Systems
4-1 Exponents
Additional Example 3: Using the Order of Operations
4(7) + 16
Substitute 4 for x, 2 for y, and 3 for z.
Evaluate the exponent.
Subtract inside the parentheses.
Multiply from left to right.
4(24 – 32) + 42
4(16 – 9) + 16
28 + 16
Evaluate x(yx – zy) + x for x = 4, y = 2, and z = 3.
y
x(yx – zy) + xy
Add. 44
4-1 Exponents
Check It Out: Example 3
60 – 7(7)
Substitute 5 for x, 2 for y, and 60 for z.
Evaluate the exponent.
Subtract inside the parentheses.
Multiply from left to right.
60 – 7(25 – 52)
60 – 7(32 – 25)
60 – 49
Evaluate z – 7(2x – xy) for x = 5, y = 2, and z = 60.
z – 7(2x – xy)
Subtract. 11
4-1 Exponents
Subtract inside the parentheses.
Evaluate 5 – (6 – 4)–3 + (– 2)0.
Evaluate the exponents.
Additional Example 3: Using the Order of Operations
5 – (6 – 4)–3 + (–2)0
= 5 – (2)–3 + (–2)0
= 5 – + 1 1 8
= 5 7 8
Add and subtract from left to right.
4-1 Exponents
Subtract inside the parentheses.
Evaluate 3 + (7 – 4)–2 + (–8)0.
Evaluate the exponents.
Check It Out: Example 3
3 + (7 – 4)–2 + (–8)0
= 3 + (3)–2 + (–8)0
= 3 + + 1 1 9
= 4 1 9
Add.
4-3 Properties of Exponents
Problem of the Day
Calculate 6 to the fourth power minus 56.
1240