Appendix A — Answer Key, Chapter 7: Exponents, Perfect Squares, and Powers of Ten

SOL 6.NS.3 · Covers textbook Chapter 7 and the companion workbook. Item numbers match the textbook; workbook items that repeat textbook problems share the same answers, and workbook-only items are keyed at the end. Reasoning answers show an acceptable response, not the only wording.

Lesson 7.1 — Exponents and Repeated Multiplication

Guided practice

1. $9^2 = 81$
2. $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$
3. $10^3 = 1{,}000$
4. $1^8 = 1$ and $8^1 = 8$
5. They are equal: $2^4 = 16$ and $4^2 = 16$.

Independent practice

6. a) $8$ b) $125$ c) $49$ d) $144$
7. a) $4^5$ b) $11^2$
8. $6^4 = 6 \times 6 \times 6 \times 6 = 1{,}296$
9. $3^3 > 4^2$, because $27 > 16$.
10. $4^2 = 16$, $5^2 = 25$, $3^3 = 27$, $2^5 = 32$ → $4^2$, $5^2$, $3^3$, $2^5$
11. $2^6 = 64$ bacteria
12. In $3^4$ the number 3 is used as a factor four times, giving $81$. In $4^3$ the number 4 is used as a factor three times, giving $64$. The base says which number is multiplied and the exponent says how many times, so switching them changes both the factor and the count.

Exit ticket 7.1

1. $64$
2. $8^3$
3. $2^5$ is greater: $2^5 = 32$ and $5^2 = 25$.
4. The base is $7$ and the exponent is $3$. The base is the number used as a factor; the exponent says it is used 3 times. So $7^3 = 7 \times 7 \times 7 = 343$.

Lesson 7.2 — Patterns with Bases and Exponents

Guided practice

1. $4$, $16$, $64$, $256$
2. Each power is the one before it multiplied by 2. Starting from $2^1 = 2$: $4$, $8$, $16$, $32$, $64$, so $2^6 = 64$.
3. $3^5 = 81 \times 3 = 243$
4. $6^4 = 1{,}296$, so the ones digit is $6$. (Every power of 6 ends in 6.)
5. Divide by the base: $1{,}000 \div 10 = 100$, so $10^2 = 100$.

Independent practice

6. $5^1 = 5$, $5^2 = 25$, $5^3 = 125$, $5^4 = 625$
7. $2^9 = 256 \times 2 = 512$; $2^7 = 256 \div 2 = 128$
8. Ones digits: $2, 4, 8, 6, 2, 4, 8, 6$. The cycle has length 4, and $12$ is a multiple of 4, so $2^{12}$ ends in the fourth digit of the cycle, $6$. (Check: $2^{12} = 4{,}096$.)
9. $3^5 = 243$ and $5^3 = 125$, so $3^5$ is larger.
10. $4^2 = 16$, $4^1 = 4$, $4^0 = 1$
11. $64 = 2^6$, so 6 rounds are played.
12. Every power of 5 is the previous power times 5. A number ending in 5 multiplied by 5 always ends in 5, because $5 \times 5 = 25$ contributes a 5 in the ones place, so the pattern never breaks.

Exit ticket 7.2

1. $3$, $9$, $27$, $81$
2. $2^{11} = 1{,}024 \times 2 = 2{,}048$
3. $9^3 = 729$, so the ones digit is $9$.
4. Multiply the value of the current power by the base. For example, $3^4 = 81$, so $3^5 = 81 \times 3 = 243$.

Lesson 7.3 — Perfect Squares

Guided practice

1. $1$, $4$, $9$, $16$, $25$
2. Yes. $7 \times 7 = 49$, so 49 tiles form a 7-by-7 square array.
3. No. $49 = 7^2$ and $64 = 8^2$, and $49 < 60 < 64$. There is no whole number between 7 and 8, so 60 is not a perfect square.
4. $13^2 = 169$
5. A 6-by-6 array of 36 tiles: 6 rows of 6, with a side length of 6.

Independent practice

6. a) $64$ b) $225$ c) $361$ d) $400$
7. Perfect squares: $81 = 9^2$, $100 = 10^2$, $121 = 11^2$. Not perfect squares: $90$ (between $81$ and $100$), $110$ (between $100$ and $121$).
8. Yes. $18 \times 18 = 324$, so 324 tiles form an 18-by-18 array.
9. No. $15^2 = 225$ and $16^2 = 256$, and $225 < 250 < 256$. No whole number lies between 15 and 16.
10. $121$, $144$, $169$, $196$
11. $13^2 = 169$, so 13 tiles run along each side. The border is the outer ring: $4 \times 13 - 4 = 48$ tiles, subtracting 4 because each corner tile would otherwise be counted twice.
12. $14^2 = 196$ and $15^2 = 225$, and $196 < 200 < 225$. A whole number squaring to 200 would have to be between 14 and 15, and there is no such whole number.

Exit ticket 7.3

1. $256$
2. Yes. $14 \times 14 = 196$.
3. No. $17^2 = 289$ and $18^2 = 324$, and $289 < 300 < 324$, so no whole number squares to 300.
4. If the tiles can be arranged into a square with the same number of rows as columns and none left over, the number is a perfect square, and the side length is the number that was squared. If any arrangement leaves an incomplete row or column, it is not.

Lesson 7.4 — Powers of Ten and Place Value

Guided practice

1. $10^4 = 10{,}000$
2. $10^5$
3. Seven zeros; $10^7 = 10{,}000{,}000$
4. $6 \times 10^2 = 600$
5. $512 = 5 \times 10^2 + 1 \times 10^1 + 2 \times 10^0$

Independent practice

6. a) $100$ b) $100{,}000$ c) $1$ d) $1{,}000{,}000$
7. a) $10^3$ b) $10^7$
8. $8{,}304 = 8 \times 10^3 + 3 \times 10^2 + 0 \times 10^1 + 4 \times 10^0$
9. a) $450{,}000$ b) $7{,}000{,}000$
10. $4^{10}$ is greater. $10^4 = 10{,}000$ and $4^{10} = 1{,}048{,}576$.
11. $5 \times 10^4 = 50{,}000$ people. Since $10^5 = 100{,}000$, the stadium holds half of $10^5$.
12. Each factor of 10 shifts every digit one place to the left and puts a zero in the ones place. Multiplying by 10 a total of $n$ times therefore adds $n$ zeros, so the exponent counts the zeros.

Exit ticket 7.4

1. $1{,}000$
2. $10^6$
3. $2{,}750 = 2 \times 10^3 + 7 \times 10^2 + 5 \times 10^1 + 0 \times 10^0$
4. Each column of the chart is worth 10 times the column to its right, so the powers increase by one exponent as you move left: ones is $10^0$, tens is $10^1$, hundreds is $10^2$, and so on. Moving one place left multiplies the place value by 10.

Chapter 7 Review

Part A — Patterns with bases and exponents (6.NS.3a)

1. a) $64$ b) $81$ c) $125$
2. $7^4 = 2{,}401$
3. $2^8 = 128 \times 2 = 256$
4. $4^2 = 16$, $5^2 = 25$, $3^3 = 27$, $2^5 = 32$ → $4^2$, $5^2$, $3^3$, $2^5$

Part B — Perfect squares with models (6.NS.3b)

5. $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$
6. $12$ tiles, since $12 \times 12 = 144$.
7. The gap is $25 - 16 = 9$. Growing a 4-by-4 array into a 5-by-5 array adds a row of 4, a column of 4, and 1 corner tile: $4 + 4 + 1 = 9$.

Part C — Justifying perfect squares (6.NS.3c)

8. Yes. $17 \times 17 = 289$.
9. No. $11^2 = 121$ and $12^2 = 144$, and $121 < 130 < 144$, so no whole number squares to 130.
10. $324 = 18^2$ and $361 = 19^2$. ($400 = 20^2$ is not between 300 and 400.)

Part D — Powers of ten and place value (6.NS.3d)

11. $10^4 = 10{,}000$ and $10^0 = 1$
12. $3{,}206 = 3 \times 10^3 + 2 \times 10^2 + 0 \times 10^1 + 6 \times 10^0$
13. $62{,}000$

Part E — Mixed application and reasoning

14. $3^6$ is greater. $6^3 = 216$ and $3^6 = 729$.
15. Side length $15$ feet, since $15^2 = 225$. Perimeter $4 \times 15 = 60$ feet.
16. No. Powers of 10 with even exponents are perfect squares: $10^0 = 1 = 1^2$, $10^2 = 100 = 10^2$, $10^4 = 10{,}000 = 100^2$. Powers with odd exponents are not: $10^1 = 10$ falls between $9 = 3^2$ and $16 = 4^2$, and $10^3 = 1{,}000$ falls between $961 = 31^2$ and $1{,}024 = 32^2$.

Workbook-only items

Page 2, table. $2^4 = 2 \times 2 \times 2 \times 2 = 16$; $5^2 = 5 \times 5 = 25$; $3^3 = 3 \times 3 \times 3 = 27$; $10^2 = 10 \times 10 = 100$; $6^2 = 6 \times 6 = 36$; $2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$

Page 2, true or false. False. $3^4 = 81$.

Page 3, evaluate. a) $8$ b) $9$ c) $16$ d) $125$ e) $49$ f) $10{,}000$ g) $1$ h) $9$

Page 3, compare. a) $2^4 = 4^2$ (both 16) b) $3^3 > 4^2$ (27 and 16) c) $5^2 < 2^5$ (25 and 32) d) $6^2 < 3^4$ (36 and 81)

Page 3, apply it. $2^6$, which equals $64$.

Page 3, explain. $3^4$ uses 3 as a factor four times and equals 81; $4^3$ uses 4 as a factor three times and equals 64. The base and the exponent play different roles, so swapping them changes the value.

Page 5, power tables. $2^n$: $2, 4, 8, 16, 32, 64, 128, 256$. $3^n$: $3, 9, 27, 81, 243$. $4^n$: $4, 16, 64, 256$. $5^n$: $5, 25, 125, 625$.

Page 5, the rule. To move one exponent up, multiply by the base. To move one exponent down, divide by the base.

Page 6, ones digits. $2, 4, 8, 6, 2, 4, 8, 6$. The cycle repeats every 4 powers. Ones digit of $2^{12}$: $6$.

Page 6, patterns downward. $4^2 = 16$, $4^1 = 4$, $4^0 = 1$. $10^2 = 100$, $10^1 = 10$, $10^0 = 1$.

Page 6, explain. Each power of 5 is the previous one times 5, and any number ending in 5 times 5 ends in 5.

Page 6, apply it. $64 = 2^6$, so there are 6 rounds.

Page 8, perfect square tables. $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ and $121, 144, 169, 196, 225, 256, 289, 324, 361, 400$

Page 8, gaps. $3$, $5$, $7$, $9$, $11$. The gaps are the consecutive odd numbers.

Page 9, justify table. $64$: yes, $8^2$. $75$: no, between $64 = 8^2$ and $81 = 9^2$. $121$: yes, $11^2$. $200$: no, between $196 = 14^2$ and $225 = 15^2$. $361$: yes, $19^2$. $350$: no, between $324 = 18^2$ and $361 = 19^2$.

Page 9, list. $121$, $144$, $169$, $196$

Page 9, apply it. 13 tiles along one side; $4 \times 13 - 4 = 48$ border tiles.

Page 11, powers of ten table. Values: $1$; $10$; $100$; $1{,}000$; $10{,}000$; $100{,}000$; $1{,}000{,}000$. Zeros: $0$; $1$; $2$; $3$; $4$; $5$; $6$.

Page 11, write as a power of ten. $1{,}000 = 10^3$; $10{,}000{,}000 = 10^7$; $100 = 10^2$; $1 = 10^0$

Page 11, multiply. $8 \times 10^3 = 8{,}000$; $34 \times 10^2 = 3{,}400$; $5 \times 10^5 = 500{,}000$

Page 12, expanded form. 1. $8 \times 10^3 + 3 \times 10^2 + 0 \times 10^1 + 4 \times 10^0$ 2. $5 \times 10^3 + 1 \times 10^2 + 4 \times 10^1 + 0 \times 10^0$ 3. $9 \times 10^2 + 0 \times 10^1 + 6 \times 10^0$ 4. $7 \times 10^4 + 2 \times 10^3 + 0 \times 10^2 + 0 \times 10^1 + 0 \times 10^0$

Page 12, compare. $4^{10}$ is greater: $10^4 = 10{,}000$ and $4^{10} = 1{,}048{,}576$.

Page 12, apply it. $50{,}000$ people, which is half of $10^5$.

Page 12, explain. Each factor of 10 shifts every digit one place left and adds a zero in the ones place, so $n$ factors of 10 produce $n$ zeros.