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(Integer)How many digits does \(2^{17}\) * \(5^{10}\) have?

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by Max@Math Revolution » Tue Sep 01, 2020 1:54 am

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(Integer) How many digits does \(2^{17}\) * \(5^{10}\) have?


A. 11
B. 13
C. 15
D. 17
E. 19
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Re: (Integer)How many digits does \(2^{17}\) * \(5^{10}\) have?

by Max@Math Revolution » Mon Sep 14, 2020 5:17 am
MathRevolution wrote:How many digits does \(2^{17}\ \) * \(5^{10}\ \) have?


A. 11
B. 13
C. 15
D. 17
E. 19

Solution:

To find the number of digits, pair up 2 and 5 with common powers, and we get

=> \(2^{17}\ \) * \(5^{10}\ \) = \(2^{7}\ \) * \(2^{10}\ \) * \(5^{10}\ \)

=> If we use the exponent property of \(a^x\ \) *\(b^x\ \) = \({ab}^x\ \) , we get

=> \(2^{17}\ \) * \(5^{10}\ \) = \(2^{7}\ \) * \(2^{10}\ \) * \(5^{10}\ \) = 128 * \(10^{10}\ \)

=> Since 128 * \(10^{10}\ \) has 128 as the first 3 digits followed by 10 zeros, the total number of digits of \(2^{17}\ \) * \(5^{10}\ \) = 3 + 10 = 13.

Therefore, B is the correct answer.

Answer B
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