Problem 7 · 2014 Math Kangaroo
Medium
Number Theory
place-value
How many digits does the result of the calculation \((2^{22})^5\times(5^{55})^2\) have?
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Answer: E — 111
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Hint 1 of 2
Pair up the 2's with the 5's to make 10's.
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Hint 2 of 2
A power of 10 has a digit count you can read straight off the exponent.
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Approach: combine into a single power of 10
- (2^{22})^5 = 2^{110} and (5^{55})^2 = 5^{110}.
- Multiplying: 2^{110}·5^{110} = 10^{110}.
- 10^{110} is 1 followed by 110 zeros, which has 111 digits.
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