The number $12^{10}\cdot 6^{-8}$ is an integer. How many digits does it have?

We are tasked with determining how many digits the integer $ 12^{10} \cdot 6^{-8} $ has.

### Step 1: Rewrite the expression
We can express $ 12 $ and $ 6 $ in terms of their prime factorizations:
\[
12 = 2^2 \cdot 3, \quad 6 = 2 \cdot 3
\]
Now, substitute these factorizations into the given expression $ 12^{10} \cdot 6^{-8} $:
\[
12^{10} = (2^2 \cdot 3)^{10} = 2^{20} \cdot 3^{10}
\]
\[
6^{-8} = (2 \cdot 3)^{-8} = 2^{-8} \cdot 3^{-8}
\]
Thus, the expression becomes:
\[
12^{10} \cdot 6^{-8} = (2^{20} \cdot 3^{10}) \cdot (2^{-8} \cdot 3^{-8})
\]
Simplifying by combining powers of 2 and powers of 3:
\[
= 2^{20 - 8} \cdot 3^{10 - 8} = 2^{12} \cdot 3^2
\]

### Step 2: Calculate the value of $ 2^{12} \cdot 3^2 $
First, calculate $ 2^{12} $ and $ 3^2 $:
\[
2^{12} = 4096, \quad 3^2 = 9
\]
Now, multiply these values:
\[
2^{12} \cdot 3^2 = 4096 \cdot 9 = 36864
\]

### Step 3: Determine the number of digits
To find how many digits $ 36864 $ has, we use the fact that the number of digits of a number $ n $ is given by:
\[
\text{Number of digits} = \lfloor \log_{10} n \rfloor + 1
\]
Now, calculate $ \log_{10} 36864 $:
\[
\log_{10} 36864 \approx 4.566
\]
Thus, the number of digits is:
\[
\lfloor 4.566 \rfloor + 1 = 4 + 1 = 5
\]

### Final Answer:
The number $ 12^{10} \cdot 6^{-8} $ has **5 digits**.

Would you like any further details or clarifications?

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Here are 5 related questions to expand your understanding:
1. How do we find the number of digits of a very large number?
2. What is the prime factorization of numbers like 12 and 6, and how does it help in simplifying expressions?
3. How can you use logarithms to quickly estimate large number properties?
4. How does the change of base formula for logarithms work in cases like this?
5. How would the number of digits change if the exponents in the original problem were larger?

**Tip:** When working with powers and prime factorizations, breaking numbers down into their prime components can simplify many calculations.

The number $12^{10} \cdot 6^{-8}$ is an integer. How many digits does it have?

Find out how to calculate the number of digits in the expression 12^10 * 6^-8 using prime factorization and logarithmic estimation. This detailed solution covers exponentiation and logarithmic techniques suitable for high school students in Grades 10-12.

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