Problem 18 · 2018 Math Kangaroo
Hard
Algebra & Patterns
substitution
How often does the summand \(2018^2\) appear under the root if the following is correct? \[\sqrt{2018^2 + 2018^2 + \cdots + 2018^2} = 2018^{10}\]
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Answer: E — \(2018^{18}\)
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Hint 1 of 2
If the summand appears k times, the inside is \(k\cdot 2018^2\).
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Hint 2 of 2
Squaring both sides gives \(k\cdot 2018^2 = 2018^{20}\).
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Approach: square both sides and solve for the count
- If \(2018^2\) appears k times, the expression is \(\sqrt{k\cdot 2018^2}\).
- Squaring the equation: \(k\cdot 2018^2 = (2018^{10})^2 = 2018^{20}\).
- So \(k = 2018^{20}\div 2018^2 = \) \(2018^{18}\).
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