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Chapter 3: Problem 7

How many digits does \(6^{700} \cdot 23^{1000}\) have?

Short Answer

Expert verified
The number \(6^{700} \cdot 23^{1000}\) has 3500 digits.

Step by step solution

01

Simplify the given expression using logarithmic properties

We have the expression \(6^{700} \cdot 23^{1000}\). Using logarithmic properties, we can multiply the exponents by taking the logarithm to the base 10 of the expression: \(\log_{10}(6^{700} \cdot 23^{1000}) = 700\log_{10}(6)+ 1000\log_{10}(23)\)
02

Calculate the logarithms and sum them up

Calculate the logarithms using a calculator: \(700\log_{10}(6) \approx 493.32\) and \(1000\log_{10}(23) \approx 3006.23\) Now, sum them up: \(493.32 + 3006.23 \approx 3499.55\)
03

Find the number of digits

Since we know the logarithm to the base 10 of the expression, we can use the formula for finding the number of digits in a number: Number of Digits = \(\lfloor \log_{10}(n) \rfloor + 1\) In our case, the number of digits is: Number of Digits = \(\lfloor 3499.55\rfloor + 1 = 3499 + 1 = 3500\) Therefore, the number \(6^{700} \cdot 23^{1000}\) has 3500 digits.

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