Problem 7 · 2024 Math Kangaroo
Medium
Arithmetic & Operations
number-systems
Which of the following expressions has the same value as \(16^{15}+16^{15}+16^{15}+16^{15}\)?
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Answer: A — \(4^{31}\)
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Hint 1 of 2
Adding four equal copies of a number is the same as multiplying that number by 4.
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Hint 2 of 2
Write everything as a power of 4: 16 = 4², so 16¹⁵ = 4³⁰, then multiply by the extra 4.
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Approach: rewrite as a single power of 4
- 16¹⁵ + 16¹⁵ + 16¹⁵ + 16¹⁵ = 4 × 16¹⁵.
- Since 16 = 4², we have 16¹⁵ = 4³⁰.
- So the sum is 4 × 4³⁰ = 4³¹.
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