Patterns and Exponents Lesson

The numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, … form a pattern. What is the rule for this pattern? Answer

This list of numbers results from finding powers of 2 in sequence. Look at the table below and you will see several patterns.

Exponential Form	Factor Form	Standard Form
2⁰ =	Any number (except 0) raised to the zero power is always equal to 1.	1
2¹ =	Any number raised to the first power is always equal to itself.	2
2² =	2 x 2 =	4
2³ =	2 x 2 x 2 =	8
2⁴ =	2 x 2 x 2 x 2 =	16
2⁵ =	2 x 2 x 2 x 2 x 2 =	32
2⁶ =	2 x 2 x 2 x 2 x 2 x 2 =	64
2⁷ =	2 x 2 x 2 x 2 x 2 x 2 x 2 =	128
2⁸ =	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =	256

Can you predict the next two numbers in the list after 256? Answer

Example 1: Rewrite the numbers 1, 3, 9, 27, 81, 243, … as a powers of 3.

Solution:

Exponential Form	Standard Form
3⁰ =	1
3¹ =	3
3² =	9
3³ =	27
3⁴ =	81
3⁵ =	243

Can you predict the next two numbers in the list after 243? Answer

Example 2: If 7³ = 343, then find 7⁴ with only one multiplication.

Solution: 7⁴ = 7³ times 7

7⁴ = 343 x 7

7⁴ = 2,401

Example 3: If 4⁵ = 1,024, then find 4⁶ with only one multiplication.

Solution: 4⁶ = 4⁵ times 4

4⁶ = 1,024 x 4

4⁶ = 4,096

Example 4: If 10⁰ = 1, and 10¹ = 10, and 10² = 100, and 10³ = 1,000, then predict the values of 10⁶ and 10⁸ in standard form.

Solution:

Exponential Form	Standard Form	Description
10⁰ =	1	The exponent is 0; the number 1 has no zeros.
10¹ =	10	The exponent is 1; the number 10 has 1 zero.
10² =	100	The exponent is 2; the number 100 has 2 zeros.
10³ =	1,000	The exponent is 3; the number 1,000 has 3 zeros.
10⁶ =	1,000,000	The exponent is 6; the number 1,000,000 has 6 zeros.
10⁸ =	100,000,000	The exponent is 8; the number 100,000,000 has 8 zeros.

Summary: When you find powers of a number in sequence, the resulting list of products forms a pattern. By examining this pattern, we can predict the next product in the list. Given the standard form of a number raised to the n^th power, we can find the standard form of that number raised to the n+1 power with a single multiplication. When you find powers of 10 in sequence, a pattern of zeros is formed in the resulting list of products.

Exercises

Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. Do not include commas in your answers. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1.	The numbers 1, 5, 25, 125, 625, … are each powers of what number? ANSWER BOX: RESULTS BOX:

2.	In Exercise 1, what is the next number in the list? ANSWER BOX: RESULTS BOX:

3.	The numbers 1, 6, 36, 216, 1296, … are each powers of what number? ANSWER BOX: RESULTS BOX:

4.	10,000,000,000,000 is 10 raised to what power? ANSWER BOX: RESULTS BOX:

5.	If 1⁴ is equal to 1, then what is 1¹⁰⁰? ANSWER BOX: RESULTS BOX: