Question

1-1 numbers and patterns
identify a possible pattern. use the pattern to write the next three numbers or figures.

1. 8, 15, 22, 29, …
2. 79, 66, 53, 40, …
3. images of figures: squares with colored squares, circles

1-2 exponents
find each value.

1. ( 8^4 )
2. ( 7^3 )
3. ( 4^5 )
4. ( 6^2 )
5. the number of bacteria in a sample doubles every hour. how many bacteria cells will there be after 8 hours if there is one cell at the beginning? write your answer as a power.

Explanation:

Step1: Find the common difference

Subtract consecutive terms: \( 15 - 8 = 7 \), \( 22 - 15 = 7 \), \( 29 - 22 = 7 \). The pattern is adding 7.

Step2: Calculate next three terms

- 5th term: \( 29 + 7 = 36 \)
- 6th term: \( 36 + 7 = 43 \)
- 7th term: \( 43 + 7 = 50 \)

Problem 2: 79, 66, 53, 40, …

Explanation:

Step1: Find the common difference

Subtract consecutive terms: \( 66 - 79 = -13 \), \( 53 - 66 = -13 \), \( 40 - 53 = -13 \). The pattern is subtracting 13.

Step2: Calculate next three terms

- 5th term: \( 40 - 13 = 27 \)
- 6th term: \( 27 - 13 = 14 \)
- 7th term: \( 14 - 13 = 1 \)

Problem 4: \( 8^4 \)

Explanation:

Step1: Expand the exponent

\( 8^4 = 8 \times 8 \times 8 \times 8 \)

Step2: Multiply step - by - step

\( 8\times8 = 64 \), \( 64\times8 = 512 \), \( 512\times8 = 4096 \)

Problem 5: \( 7^3 \)

Explanation:

Step1: Expand the exponent

\( 7^3 = 7\times7\times7 \)

Step2: Multiply step - by - step

\( 7\times7 = 49 \), \( 49\times7 = 343 \)

Problem 6: \( 4^5 \)

Explanation:

Step1: Expand the exponent

\( 4^5 = 4\times4\times4\times4\times4 \)

Step2: Multiply step - by - step

\( 4\times4 = 16 \), \( 16\times4 = 64 \), \( 64\times4 = 256 \), \( 256\times4 = 1024 \)

Problem 7: \( 6^2 \)

Explanation:

Step1: Expand the exponent

\( 6^2 = 6\times6 \)

Step2: Multiply

\( 6\times6 = 36 \)

Problem 8: Bacteria Growth

Explanation:

Step1: Identify the pattern

The number of bacteria doubles every hour. Starting with 1 cell, after 1 hour: \( 2^1 \), after 2 hours: \( 2^2 \),..., after \( n \) hours: \( 2^n \).

Step2: Apply for 8 hours

After 8 hours, the number of bacteria is \( 2^8 \).

Answers:

1. Next three numbers: \( \boldsymbol{36, 43, 50} \)
2. Next three numbers: \( \boldsymbol{27, 14, 1} \)
4. \( 8^4=\boldsymbol{4096} \)
5. \( 7^3=\boldsymbol{343} \)
6. \( 4^5=\boldsymbol{1024} \)
7. \( 6^2=\boldsymbol{36} \)
8. Number of bacteria: \( \boldsymbol{2^8} \) (or 256, but as a power, \( 2^8 \))

Step1: Find the common difference

Subtract consecutive terms: \( 15 - 8 = 7 \), \( 22 - 15 = 7 \), \( 29 - 22 = 7 \). The pattern is adding 7.

Step2: Calculate next three terms

• 5th term: \( 29 + 7 = 36 \)
• 6th term: \( 36 + 7 = 43 \)
• 7th term: \( 43 + 7 = 50 \)

Problem 2: 79, 66, 53, 40, …

Step1: Find the common difference

Subtract consecutive terms: \( 66 - 79 = -13 \), \( 53 - 66 = -13 \), \( 40 - 53 = -13 \). The pattern is subtracting 13.

Step2: Calculate next three terms

• 5th term: \( 40 - 13 = 27 \)
• 6th term: \( 27 - 13 = 14 \)
• 7th term: \( 14 - 13 = 1 \)

Problem 4: \( 8^4 \)

Step1: Expand the exponent

\( 8^4 = 8 \times 8 \times 8 \times 8 \)

Step2: Multiply step - by - step

\( 8\times8 = 64 \), \( 64\times8 = 512 \), \( 512\times8 = 4096 \)

Problem 5: \( 7^3 \)

Answer:

s:

1. Next three numbers: \( \boldsymbol{36, 43, 50} \)
2. Next three numbers: \( \boldsymbol{27, 14, 1} \)
3. \( 8^4=\boldsymbol{4096} \)
4. \( 7^3=\boldsymbol{343} \)
5. \( 4^5=\boldsymbol{1024} \)
6. \( 6^2=\boldsymbol{36} \)
7. Number of bacteria: \( \boldsymbol{2^8} \) (or 256, but as a power, \( 2^8 \))