Let’s solve each problem step by step. We need to express each product as a single exponent of the given base number.
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Problem 1: 3 × 9 × 81 × 729 as an exponent of 3
First, write each number as a power of 3:
- 3 = 3¹
- 9 = 3²
- 81 = 3⁴ (because 3×3×3×3 = 81)
- 729 = 3⁶ (because 3⁶ = 729 — check: 3=243, 3=729)
Now multiply them:
3¹ × 3² × 3⁴ × 3⁶ = 3^(1+2+4+6) = 3¹³
✔ Answer for #1:
3¹³
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Problem 2: 16 × 64 × 256 × 32 as an exponent of 2
Write each as power of 2:
- 16 = 2⁴
- 64 = 2⁶
- 256 = 2⁸
- 32 = 2⁵
Multiply: 2⁴ × 2⁶ × 2⁸ × 2⁵ = 2^(4+6+8+5) = 2²³
✔ Answer for #2:
2²³
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Problem 3: 25 × 625 × 15625 × 125 as an exponent of 5
Powers of 5:
- 25 = 5²
- 625 = 5 (5×5×5×5 = 625)
- 15625 = 5⁶? Let’s check:
5⁵ = 3125
5⁶ = 15625 → yes!
- 125 = 5³
So: 5² × 5⁴ × 5⁶ × 5³ = 5^(2+4+6+3) = 5¹
✔ Answer for #3:
5¹
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Problem 4: 64 × 256 × 1024 × 4096 as an exponent of 4
Wait — base is 4, not 2. So we must write each number as a power of 4.
But note: 4 = 2², so powers of 4 are even powers of 2.
Alternatively, convert each to base 4 directly:
- 64 = 4³? Check: 4¹=4, 4²=16, 4³=64 → yes
- 256 = 4⁴? 4 = 256 → yes
- 1024 = ? 4⁵ = 1024? 4⁵ = 4×4×4×4×4 = 1024 → yes
- 4096 = 4⁶? 4 = 4096 → yes (since 4⁵=1024, 4⁶=4096)
So: 4³ × 4⁴ × 4⁵ × 4⁶ = 4^(3+4+5+6) = 4¹⁸
✔ Answer for #4:
4¹⁸
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Problem 5: 10 × 1000 × 100 × 10000 as an exponent of 10
Easy! These are all powers of 10:
- 10 = 10¹
- 1000 = 10³
- 100 = 10²
- 10000 = 10⁴
Multiply: 10¹ × 10³ × 10² × 10⁴ = 10^(1+3+2+4) = 10¹⁰
✔ Answer for #5:
10¹⁰
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Problem 6: 64 × 4096 × 512 × 32768 as an exponent of 8
Base is 8. Write each as power of 8.
Note: 8 = 2³, but let’s try direct powers of 8.
- 64 = 8²? 8×8=64 → yes
- 4096 = 8⁴? Let’s check:
8¹=8
8²=64
8³=512
8=4096 → yes
- 512 = 8³ → from above
- 32768 = ? 8⁵ = 32768?
8⁴=4096, 8⁵=4096×8=32768 → yes
So: 8² × 8⁴ × 8³ × 8⁵ = 8^(2+4+3+5) = 8¹⁴
✔ Answer for #6:
8¹⁴
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Final Answer:
1) 3¹³
2) 2²³
3) 5¹
4) 4¹⁸
5) 10¹⁰
6) 8¹⁴
Parent Tip: Review the logic above to help your child master the concept of multiply like bases worksheet.
