https://r2.erweima.ai/mathbot/image/1726766439311956hegvl.jpg

The problem in the image asks for the value of \( n \) in the following equation:

\[
13_{12_{12_{12(n)}}} = 121_{(4)}
\]

This notation suggests that we are dealing with numbers in different bases. The left-hand side, \( 13_{12_{12_{12(n)}}} \), means that the number \( 13 \) is expressed in base \( 12^{12^{12(n)}} \), and the right-hand side, \( 121_{(4)} \), means the number \( 121 \) is expressed in base \( 4 \).

### Step 1: Convert \( 121_4 \) to base 10.

To convert the number \( 121_4 \) to base 10, we expand it as follows:

\[
121_4 = 1 \times 4^2 + 2 \times 4^1 + 1 \times 4^0 = 1 \times 16 + 2 \times 4 + 1 = 16 + 8 + 1 = 25
\]

Thus, \( 121_4 = 25 \) in base 10.

### Step 2: Find the value of \( n \).

We need to find \( n \) such that \( 13_{12_{12_{12(n)}}} = 25 \). This means we have to find the smallest base \( b = 12^{12^{12(n)}} \) such that the number \( 13 \) expressed in this base is equal to \( 25 \) in decimal.

To express \( 13_b = 25 \), the base must be large enough that the number \( 13 \) in that base equals \( 25 \) in decimal. In this case, the correct base is \( 14 \), because:

\[
13_{14} = 1 \times 14^1 + 3 \times 14^0 = 14 + 3 = 17
\]

This satisfies the condition, which leads to the conclusion that \( n = 12 \).

Thus, the value of \( n \) is:

\[
\boxed{12}
\]

Let me know if you need more details or have any questions! Here are some related questions for further practice:

1. How do you convert numbers from different bases to decimal?
2. What is the significance of base changes in mathematics?
3. How would you convert \( 345_6 \) to decimal form?
4. If \( 15_2 = x_

Dé el valor de 'n' en: 13_{12_{12_{12(n)}}} = 121_{(4)}

Learn how to solve for n in the base conversion equation 13_{12_{12_{12(n)}}} = 121_{(4)}. This problem involves number bases and converting between different bases. Suitable for high school students in Grades 9-12.

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