/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 How many digits does \(5^{999} \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 8

How many digits does \(5^{999} \cdot 17^{2222}\) have?

Short Answer

Expert verified
The expression \(5^{999} \cdot 17^{2222}\) has 3149 digits.

Step by step solution

01

Use logarithm properties to simplify the expression

Given the expression: \(5^{999} \cdot 17^{2222}\), we can take the logarithm of both sides. Remember that logarithm is an operation that allows us to make exponentiation problems easier to handle. Considering the properties of logarithms, we can rewrite the expression as: \(log(5^{999} \cdot 17^{2222}) = 999\cdot log(5) + 2222\cdot log(17)\)
02

Calculate the value of the simplified expression

Now we need to find the value of the simplified expression. Using a calculator or a logarithm table, we can find the value of \(log(5)\) and \(log(17)\): \(log(5) \approx 0.69897\) \(log(17) \approx 1.23045\) Now substitute these values into the expression and perform the calculation: \(999 \cdot 0.69897 + 2222 \cdot 1.23045 \approx 3148.128\)
03

Find the number of digits using the characteristic of the logarithm

The number of digits in a given value can be determined by finding the characteristic of its logarithm. The characteristic is the integer part of the logarithm. In our case, the integer part of \(3148.128\) is 3148. However, the logarithm we calculated in Step 2 represents the logarithm of the given expression, not the number of digits in the expression. To find the number of digits, we need to add 1 to the characteristic i.e. Number of digits = Characteristic + 1 = 3148 + 1 = 3149
04

Final answer

Therefore, the expression \(5^{999} \cdot 17^{2222}\) has 3149 digits.

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