Students can use the **Spectrum Math Grade 8 Answer Key** **Chapter 1 Lesson 1.2 Equivalent Expressions with Exponents** as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 1 Lesson 1.2 Equivalent Expressions with Exponents Answers Key

To multiply powers with the same base, combine bases, add the exponents, then simplify.

2^{2} × 2^{3} = 2^{2 + 3} = 2^{5} = 32

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

3^{5} ÷ 3^{2} = 3^{5 – 2} = 3^{3} = 27

**Find the value of each expression.**

Question 1.

a. 7^{2} = ____

Answer: 49

A power of a number represents repeated multiplication of the number by itself.

7^{2} =7 x 7 and is read 7 to the second power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 7^{2}, 7 is the base and 2 is the exponent.

7^{2} means 7 is used as a factor 2 times.

7 x 7 = 7^{2}

7 x 7 = 49

7^{2} = 49

b. 8^{3} = ____

Answer: 512

A power of a number represents repeated multiplication of the number by itself.

8^{3} = 8 x 8 x 8 and is read 8 to the third power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 8^{3} , 8 is the base and 3 is the exponent.

8^{3} means 8 is used as a factor 3 times.

8 x 8 x 8 = 8^{3}

8 x 8 x 8 = 512

8^{3} = 512

c. 4^{3} = _____

Answer: 64

A power of a number represents repeated multiplication of the number by itself.

4^{3} = 4 x 4 x 4 and is read 4 to the third power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 4^{3}, 4 is the base and 3 is the exponent.

4^{3} means 4 is used as a factor 3 times.

4 x 4 x 4 = 4^{3}

4 x 4 x 4 = 64

4^{3} = 64

Question 2.

a. 10^{2} = ____

Answer: 100

A power of a number represents repeated multiplication of the number by itself.

10^{2} =10 x 10 and is read 10 to the second power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 10^{2} , 10 is the base and 2 is the exponent.

10^{2} means 10 is used as a factor 2 times.

10 x 10 = 10^{2}

10 x 10 = 100

10^{2} = 100

b. 9^{4} = ____

Answer: 6561

A power of a number represents repeated multiplication of the number by itself.

9^{4} = 9 x 9 x 9 x 9 and is read 9 to the fourth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 9^{4}, 9 is the base and 4 is the exponent.

9^{4} means 9 is used as a factor 4 times.

9 x 9 x 9 x 9 = 9^{4}

9 x 9 x 9 x 9 = 6561

9^{4} = 6561

c. 11^{5} = _____

Answer: 161051

A power of a number represents repeated multiplication of the number by itself.

11^{5} = 11 x 11 x 11 x 11 x 11 and is read 11 to the fifth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 11^{5}, 11 is the base and 5 is the exponent.

11^{5} means 11 is used as a factor 5 times.

11 x 11 x 11 x 11 x 11 = 11^{5}

11 x 11 x 11 x 11 x 11 = 161051

11^{5} = 161051

Question 3.

a. 17^{3} = ____

Answer: 4913

A power of a number represents repeated multiplication of the number by itself.

17^{3} = 17 x 17 x 17 and is read 17 to the third power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 17^{3}, 17 is the base and 3 is the exponent.

17^{3} means 17 is used as a factor 3 times.

17 x 17 x 17 = 17^{3}

17 x 17 x 17 = 4913

17^{3} = 4913

b. 5^{6} = ____

Answer: 15625

A power of a number represents repeated multiplication of the number by itself.

5^{6} = 5 × 5 × 5 × 5 × 5 × 5 and is read 5 to the sixth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 5^{6}, 5 is the base and 6 is the exponent.

5^{6} means 5 is used as a factor 6 times.

5 × 5 × 5 × 5 × 5 × 5 = 5^{6}

5 × 5 × 5 × 5 × 5 × 5 = 15625

5^{6} = 15625

c. 6^{4} = _____

Answer: 1296

A power of a number represents repeated multiplication of the number by itself.

6^{4} = 6 × 6 × 6 × 6 and is read 6 to the fourth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 6^{4}, 6 is the base and 4 is the exponent.

6^{4} means 6 is used as a factor 4 times.

6 × 6 × 6 × 6 = 6^{4}

6 × 6 × 6 × 6 = 1296

6^{4 }= 1296

Question 4.

a. 21^{3} = ____

Answer: 9261

A power of a number represents repeated multiplication of the number by itself.

21^{3} = 21 x 21 x 21 and is read 21 to the third power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 21^{3}, 21 is the base and 3 is the exponent.

21^{3} means 21 is used as a factor 3 times.

21 x 21 x 21 = 21^{3}

21 x 21 x 21 = 9261

21^{3} = 9261

b. 16^{4} = ____

Answer: 65536

A power of a number represents repeated multiplication of the number by itself.

16^{4} = 16 × 16 × 16 × 16 and is read 16 to the fourth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 16^{4}, 16 is the base and 4 is the exponent.

16^{4} means 16 is used as a factor 4 times.

16 × 16 × 16 × 16 = 6^{4}

16 × 16 × 16 × 16 = 65536

6^{4 }= 65536

c. 12^{5} = _____

Answer: 248832

A power of a number represents repeated multiplication of the number by itself.

12^{5} = 12 x 12 x 12 x 12 x 12 and is read 12 to the fifth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 12^{5}, 12 is the base and 5 is the exponent.

12^{5} means 11 is used as a factor 5 times.

12 x 12 x 12 x 12 x 12 = 12^{5}

12 x 12 x 12 x 12 x 12 = 248832

12^{5} = 248832

**Rewrite each expression as one base and one exponent. Then, find the value.**

Question 5.

a. 8^{2} × 8^{3} = ____

Answer: 32768

To multiply powers with the same base, combine bases, add the exponents, then simplify.

8^{2} × 8^{3} = 8^{2 + 3} = 8^{5} = 32768

b. 3^{3} × 3^{3} = ____

Answer: 729

To multiply powers with the same base, combine bases, add the exponents, then simplify.

3^{3} × 3^{3} = 3^{3 + 3} = 3^{6} = 729

c. 2^{2} × 2^{2} = ____

Answer: 16

To multiply powers with the same base, combine bases, add the exponents, then simplify.

2^{2} × 2^{2} = 2^{2 + 2} = 2^{4} = 16

Question 6.

a. 7^{4} ÷ 7^{2} = ___

Answer: 49

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

7^{4} ÷ 7^{2} = 7^{4 – 2} = 7^{2} = 49

b. 9^{5} ÷ 9^{3} = ___

Answer: 81

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

9^{5} ÷ 9^{3} = 9^{5 – 3} = 9^{2} = 81

c. 16^{4} ÷ 16^{2} = ___

Answer: 256

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

16^{4} ÷ 16^{2} = 16^{4 – 2} = 16^{2} = 256

Question 7.

a. 6^{4} × 6^{1} = ____

Answer: 7776

To multiply powers with the same base, combine bases, add the exponents, then simplify.

6^{4} × 6^{1 }= 6^{4 + 1} = 6^{5} = 7776

b. 4^{4} × 4^{2} = ____

Answer: 4096

To multiply powers with the same base, combine bases, add the exponents, then simplify.

4^{4} × 4^{2 } = 4^{4 + 2} = 4^{6} = 4096

c. 3^{2} × 3^{2} = ____

Answer: 81

To multiply powers with the same base, combine bases, add the exponents, then simplify.

3^{2} × 3^{2} = 3^{2 + 2} = 3^{4} = 81

Question 8.

a. 10^{6} ÷ 10^{4} = ____

Answer: 100

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

10^{6} ÷ 10^{4} = 10^{6 – 4} = 10^{2} = 100

b. 8^{3} ÷ 8^{2} = ____

Answer: 8

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

8^{3} ÷ 8^{2} = 8^{3 – 2} = 8^{1} = 8

c. 7^{6} ÷ 7^{3} = ____

Answer: 343

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

7^{6} ÷ 7^{3} = 7^{6 – 3} = 7^{3} = 343

Question 9.

a. 5^{3} × 5^{2} = ____

Answer: 3125

To multiply powers with the same base, combine bases, add the exponents, then simplify.

5^{3} × 5^{2} = 5^{3 + 2} = 5^{5} = 3125

b. 10^{3} × 10^{4} = ____

Answer: 10000000

To multiply powers with the same base, combine bases, add the exponents, then simplify.

10^{3} × 10^{4} = 10^{3 + 4} = 10^{7} = 10000000

c. 15^{2} × 15^{1} = ____

Answer: 3375

To multiply powers with the same base, combine bases, add the exponents, then simplify.

15^{2} × 15^{1}= 15^{2 + 1} = 15^{3} = 3375

d. 6^{6} ÷ 6^{3} = ____

Answer: 216

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

6^{6} ÷ 6^{3 }= 6^{6 – 3} = 6^{3} = 216

**Rewrite each multiplication or division expression using a base and an exponent.**

Question 1.

a. 4^{3} × 4^{5} = _____

Answer: 65536

To multiply powers with the same base, combine bases, add the exponents, then simplify.

4^{3} × 4^{5}= 4^{3 + 5} = 4^{8} = 65536

b. 9^{2} × 9^{3} = _____

Answer: 59049

To multiply powers with the same base, combine bases, add the exponents, then simplify.

9^{2} × 9^{3 }= 9^{2 + 3} = 9^{5} = 59049

Question 2.

a. (3 × 3 × 3) × (3 × 3) = _______

Answer: 3^{3} × 3^{2
}3^{3 + 2} = 3^{5 }= 243

To multiply powers with the same base, combine bases, add the exponents, then simplify.

(3 × 3 × 3) × (3 × 3) = 3^{3} × 3^{2}^{ }= 3^{3 + 2} = 3^{5} = 243

b. 5^{6} ÷ 5^{3} = _______

Answer: 125

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

5^{6} ÷ 5^{3} = 5^{6 – 3} = 5^{3} = 125

Question 3.

a. 8^{5} ÷ 8 = _______

Answer: 4096

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

8^{5} ÷ 8 = 8^{5 – 1} = 8^{4} = 4096

b. (2 × 2 × 2 × 2) ÷ (2 × 2) = _______

Answer: 2^{4} × 2^{2
}2^{4 + 2} = 2^{6 }= 64

To multiply powers with the same base, combine bases, add the exponents, then simplify.

(2 × 2 × 2 × 2) ÷ (2 × 2) = 2^{4} × 2^{2}^{ }= 2^{4 + 2 }= 2^{6} = 64

Question 4.

a. (5 × 5) × (5 × 5) = ______

Answer: 5^{2} × 5^{2
}5^{2 + 2} = 5^{4 }= 625

To multiply powers with the same base, combine bases, add the exponents, then simplify.

(5 × 5) × (5 × 5)= 5^{2} × 5^{2 }= 5^{2 + 2} = 5^{4 }= 625

b. 9^{9} ÷ 9^{5} = _______

Answer: 6561

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

9^{9} ÷ 9^{5} = 9^{9 – 5} = 9^{4} = 6561

Question 5.

a. 10^{3} × 10 = ______

Answer: 1000

To multiply powers with the same base, combine bases, add the exponents, then simplify.

10^{3} × 10 = 10^{3 + 1} = 10^{4} = 1000

b. 6^{5} ÷ 6^{2} = _______

Answer: 216

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

6^{5} ÷ 6^{2} = 6^{5 – 2} = 6^{3} = 216

Question 6.

a. 4^{3} ÷ 4^{2} = _______

Answer: 4

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

4^{3} ÷ 4^{2} = 4^{3 – 2} = 4^{1} = 4

b. (7 × 7 × 7) ÷ 7 = _______

Answer: 7^{3} ÷ 7^{1
}7^{3 – 1} = 7^{2 }= 49

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

(7 × 7 × 7) ÷ 7 =7^{3} ÷ 7^{1}= 7^{3 – 1} = 7^{2 }= 49

Question 7.

a. 11^{5} × 11^{2} = _______

Answer: 19487171

To multiply powers with the same base, combine bases, add the exponents, then simplify.

11^{5} × 11^{2} = 11^{5 + 2} = 11^{7} = 19487171

b. 6 × 6^{5} = _______

Answer: 46656

To multiply powers with the same base, combine bases, add the exponents, then simplify.

6 × 6^{5} = 6^{1 + 5} = 6^{6} = 46656

Question 8.

a. (8 × 8 × 8 × 8) ÷ (8 × 8) = _____

Answer: 8^{4} ÷ 8^{2
}8^{4- 2} = 8^{2 }= 64

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

(8 × 8 × 8 × 8) ÷ (8 × 8) =8^{4} ÷ 8^{2} =8^{4- 2} = 8^{2 }= 64

b. 5^{3} × 5^{2} = ____

Answer: 3125

To multiply powers with the same base, combine bases, add the exponents, then simplify.

5^{3} × 5^{2} = 5^{3 + 2} = 5^{5} = 3125

Question 9.

a. 12^{9} × 12^{2} = ____

Answer: 743008370688

To multiply powers with the same base, combine bases, add the exponents, then simplify.

12^{9} × 12^{2 }= 12^{9 + 2} = 12^{11} = 743008370688

b. 11^{10} ÷ 11^{4} = ____

Answer: 1771561

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

11^{10} ÷ 11^{4} = 11^{10 – 4} = 11^{6} = 1771561

Question 10.

a. 3^{4} × 3^{4} = ____

Answer: 6561

To multiply powers with the same base, combine bases, add the exponents, then simplify.

3^{4} × 3^{4} = 3^{4 + 4} = 3^{8} = 6561

b. (4 × 4 × 4 × 4) ÷ 4 = ____

Answer: 64

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

(4 × 4 × 4 × 4) ÷ 4 = 4^{4} ÷ 4^{1} = 4^{4 – 1} = 4^{3} = 64

Question 11.

a. (5 × 5 × 5) ÷ 5 = _____

Answer: 25

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

(5 × 5 × 5) ÷ 5 =5^{3} ÷ 5^{1} =5^{3- 1} = 5^{2} = 25

b. 6^{8} × 6^{4} = _____

Answer: 2176782336

To multiply powers with the same base, combine bases, add the exponents, then simplify.

6^{8} × 6^{4} = 6^{8 + 4} = 6^{12} = 2176782336

Question 12.

a. 4^{12} ÷ 4^{6} = ____

Answer: 4096

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

4^{12} ÷ 4^{6} = 4^{12- 6} = 4^{6} = 4096

b. 3^{3} × 3^{9} = ____

Answer: 531441

To multiply powers with the same base, combine bases, add the exponents, then simplify.

3^{3} × 3^{9} = 3^{3 + 9} = 3^{12} = 531441

Question 13.

a. (6 × 6 × 6 × 6) ÷ (6 × 6 × 6) = _____

Answer: 6

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

(6 × 6 × 6 × 6) ÷ (6 × 6 × 6) = 6^{4} ÷ 6^{3} = 6^{4 – 3} = 6^{1} = 6

b. 15^{8} ÷ 15^{3} = _____

Answer: 759375

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

15^{8} ÷ 15^{3} = 15^{8 – 3} = 15^{5} = 759375

Question 14.

a. 9^{9} × 9^{6} = ____

Answer: 205891132094649

To multiply powers with the same base, combine bases, add the exponents, then simplify.

9^{9} × 9^{6} = 9^{9 + 6} = 9^{15} = 205891132094649

b. 7^{8} × 7^{2} = ____

Answer: 282475249

To multiply powers with the same base, combine bases, add the exponents, then simplify.

7^{8} × 7^{2} = 7^{8 + 2} = 7^{10} = 282475249

Question 15.

a. 2^{7} ÷ 2 = ____

Answer: 64

To divide powers with the same base, combine bases, subtract the exponents, then simplify.

2^{7} ÷ 2 = 2^{7 – 1} = 2^{6} = 64

b. 4^{11} × 4 = ____

Answer: 16777216

To multiply powers with the same base, combine bases, add the exponents, then simplify.

4^{11} × 4 = 4^{11 + 1} = 4^{12} = 16777216