Question - Finding Base Value for Number Equality
Solution:
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.
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