How many digits are in 99 to the 99th power?
a) 198
b) 199
c) 200
d) 201
The Correct Answer and Explanation is:
To determine how many digits are in ( 99^{99} ), we can use the following method involving logarithms.
Step 1: Formula for the number of digits
The number of digits ( D ) in a number ( N ) is given by the formula:
[
D = \lfloor \log_{10}(N) \rfloor + 1
]
Where ( \lfloor x \rfloor ) represents the greatest integer less than or equal to ( x ), and ( \log_{10}(N) ) is the base-10 logarithm of ( N ).
Step 2: Apply the formula to ( 99^{99} )
We need to calculate ( \log_{10}(99^{99}) ). Using the logarithmic identity ( \log_{10}(a^b) = b \log_{10}(a) ), we get:
[
\log_{10}(99^{99}) = 99 \times \log_{10}(99)
]
Now, we need to calculate ( \log_{10}(99) ). We know that:
[
\log_{10}(99) \approx 1.9956
]
Thus:
[
\log_{10}(99^{99}) = 99 \times 1.9956 = 197.4644
]
Step 3: Calculate the number of digits
Now that we have the logarithm, we can find the number of digits:
[
D = \lfloor 197.4644 \rfloor + 1 = 197 + 1 = 198
]
Final Answer:
The number of digits in ( 99^{99} ) is 198.
Therefore, the correct answer is:
a) 198
Explanation Summary:
We used the logarithmic formula for determining the number of digits in a large number. By calculating the logarithm of ( 99^{99} ) using the base-10 logarithm, we found that it has 198 digits. The key steps included calculating ( \log_{10}(99) ), multiplying by 99, and applying the formula to determine the number of digits.