Finding Base Value for Number Equality
The question asks you to find the value of \( n \) such that the number \( 14 \) in base \( n \) is equal to the number \( 22 \) in base five.
To find the value of \( n \), we can convert the given number \( 22 \) in base five to base ten and then equate it to the number \( 14 \) in base \( n \), also represented in base ten.
To convert \( (22)_5 \) to base ten, use the following method:
- \( (22)_5 = 2 \cdot 5^1 + 2 \cdot 5^0 \)
- \( (22)_5 = 2 \cdot 5 + 2 \cdot 1 \)
- \( (22)_5 = 10 + 2 \)
- \( (22)_5 = 12 \) in base ten.
Now that we know \( (22)_5 \) is equal to \( 12 \) in base ten, we can set this equal to \( 14 \) in base \( n \) and also convert that to base ten. The base ten representation of \( 14 \) in base \( n \) is:
- \( 1 \cdot n^1 + 4 \cdot n^0 \)
- \( n + 4 \)
Equating this to the base ten value we found:
- \( n + 4 = 12 \)
To solve for \( n \), subtract 4 from both sides:
- \( n = 12 - 4 \)
- \( n = 8 \)
Hence, the value of \( n \) is 8.
