Find the equivalent in base 10 of the first 4-digit number in base 3.

To find the equivalent in base 10 of the first 4-digit number in base 3, we start by identifying the first 4-digit number in base 3.

In base 3, the smallest 4-digit number is \(1000_3\). Now, we'll convert this to base 10.

The number \(1000_3\) can be expanded as follows:
\[
1000_3 = 1 \cdot 3^3 + 0 \cdot 3^2 + 0 \cdot 3^1 + 0 \cdot 3^0
\]
Calculating each term:
\[
1 \cdot 3^3 = 1 \cdot 27 = 27
\]
\[
0 \cdot 3^2 = 0 \cdot 9 = 0
\]
\[
0 \cdot 3^1 = 0 \cdot 3 = 0
\]
\[
0 \cdot 3^0 = 0 \cdot 1 = 0
\]

Adding these together:
\[
27 + 0 + 0 + 0 = 27
\]

So, the equivalent in base 10 of the first 4-digit number in base 3 (\(1000_3\)) is \(27_{10}\).

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the base 10 equivalent of the 4-digit number \(2001_3\)?
2. How do you convert any base 3 number to base 10?
3. What is the largest 4-digit number in base 3, and what is its base 10 equivalent?
4. How do base conversions between non-decimal systems work in general?
5. What is the first 5-digit number in base 3, and its base 10 equivalent?
6. How does base 3 compare to binary in terms of digit length for the same number?
7. Can you find the base 3 equivalent of the number \(81_{10}\)?
8. What is the smallest 3-digit number in base 3, and its base 10 equivalent?

**Tip:** Always start conversions from the highest place value when working with non-decimal bases to avoid confusion.

Discover how to convert the first 4-digit number in base 3 to base 10 with a step-by-step explanation. This problem involves understanding number systems and base conversion. Suitable for students in Grades 6-8.

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