If 142 base n is equal to 47 base ten, what is the value of base n.
To solve this problem, we need to find the value of base n.
We are given that 142 base n is equal to 47 base ten.
Let's convert 142 base n to base ten:
142 base n = 1*n^2 + 4*n^1 + 2*n^0
Using the expanded form, we can express this equation as:
142 base n = n^2 + 4n + 2
Now, we can set this equation equal to 47 base ten:
n^2 + 4n + 2 = 47
Rearranging the equation:
n^2 + 4n + 2 - 47 = 0
n^2 + 4n - 45 = 0
Now, we can factor this equation:
(n + 9)(n - 5) = 0
Setting each factor equal to zero:
n + 9 = 0 or n - 5 = 0
n = -9 or n = 5
Since the base cannot be negative, the value of base n is 5.
To find the value of base n, we need to convert both numbers to base 10 and solve for n.
Let's start by converting 142 base n to base 10. In base n, the digits represent powers of n. So, the number 142 can be written as:
1 * n^2 + 4 * n^1 + 2 * n^0
Simplifying this expression, we get:
n^2 + 4n + 2
Now, let's convert 47 base ten to base 10. Since it is already in base 10, it can be written as:
47 * 10^0
Now, equating the two expressions and solving for n:
n^2 + 4n + 2 = 47
Rearranging the equation, we get:
n^2 + 4n + 2 - 47 = 0
n^2 + 4n - 45 = 0
Now, we can solve this quadratic equation to find the possible values of n. Factoring the equation, we get:
(n - 5)(n + 9) = 0
Setting each factor equal to zero, we have:
n - 5 = 0 or n + 9 = 0
Solving each equation, we find:
n = 5 or n = -9
Since n represents the base of the number system, it must be a positive integer. Therefore, the value of base n is 5.
To find the value of base n, we need to solve the equation 142 base n = 47 base ten.
Let's break down what this equation means:
142 base n represents a number in base n. It can be expanded as:
1 * n^2 + 4 * n^1 + 2 * n^0
= n^2 + 4n + 2
47 base ten represents a number in base ten. It can be expanded as:
4 * 10^1 + 7 * 10^0
= 40 + 7
So our equation becomes:
n^2 + 4n + 2 = 40 + 7
Simplifying, we get:
n^2 + 4n + 2 = 47
Rearranging the equation, we have:
n^2 + 4n - 45 = 0
Now, we need to solve this quadratic equation to find the value of n. We can factorize the equation as:
(n + 9)(n - 5) = 0
Setting each factor equal to zero, we have two possible solutions:
n + 9 = 0 or n - 5 = 0
If n + 9 = 0, then n = -9
If n - 5 = 0, then n = 5
However, since base n cannot be negative, the only valid solution is n = 5.
Therefore, the value of base n is 5.