Name _______________________________________
Date __________________
Hour ___________
Investigation 5.1 Looking for Patterns Among Exponents
Focus Question: What patterns did you observe in the table of powers?
The values of 𝑎 𝑥 for a given number are called powers of a. You just looked at situations involving powers of 2. In Problem 5.1, you will
explore patterns with powers of 2 and other numbers.
A. Complete the table.
x
1𝑥
2𝑥
3𝑥
4𝑥
5𝑥
6𝑥
7𝑥
8𝑥
9𝑥
10𝑥
1
1
2
3
4
5
6
7
8
9
10
2
1
4
9
16
25
36
49
64
81
100
3
1
8
27
64
216
343
729
1,000
4
1
16
81
256
5
1
32
243
6
1
64
729
7
1
2,187
8
1
256
625
2,401
4,096
6,561
10,000
16,807
32,768
59,049
100,000
262,144
531,441
3,125
7,776
4,096
15,625
46,656
16,384
78,125
279,936
823,543
65,536
390,625
1,679,616
5,764,801
2,097,152
10,000,000
16,777,216 43,046,721 100,000,000
B. Describe any patterns that you see in the rows and columns.
Growing Growing Growing
Investigation 5: Patterns with Exponents
C. Delmar noticed that if you extend the pattern upwards in the table for negative values of x, you get the values shown below.
1. Complete the table below to show how this works for columns 3𝑥 through 10𝑥 .
1
4
1
2
-2
1
-1
1
0
1
1
x
1𝑥
2𝑥
3𝑥
4𝑥
5𝑥
6𝑥
7𝑥
8𝑥
9𝑥
10𝑥
1
1
2
3
4
5
6
7
8
9
10
2
1
4
9
16
25
36
49
64
81
100
3
1
8
27
64
125
216
343
512
729
1,000
4
1
16
81
256
625
1,296
2,401
4,096
6,561
10,000
5
1
32
243
1,024
3,125
7,776
16,807
32,768
59,049
100,000
6
1
64
729
4,096
15,625
46,656
117,649
262,144
531,441
1,000,000
7
1
128
2,187
16,384
78,125
279,936
823,543
2,097,152
4,782,969
10,000,000
8
1
256
6,561
65,536
390,625
2. Delmar claims that 𝑎0 = 1 and that 𝑎 −1 =
Growing Growing Growing
1
𝑎
1,679,616 5,764,801 16,777,216 43,046,721 100,000,000
for any positive number a. Do you agree? Explain.
Investigation 5: Patterns with Exponents