Express 111 base n in base 10 and hence solve the equation 111n =37 base 10
Answer please. !!!
111 = n^2 + n + 1
now solve the quadratic.
I suspect a typo ...
To express a number in base 10, we need to convert it to the decimal system.
In order to convert 111 base n to base 10, we need to multiply each digit by the corresponding power of n and then sum them up.
So, let's express 111 base n in base 10:
111 base n = 1 * n^2 + 1 * n^1 + 1 * n^0
Now, let's solve the equation 111n = 37 base 10 by converting both sides to base 10.
Given equation: 111n = 37 base 10
To convert 111n to base 10, we need to express it as explained earlier.
111n base n = 1 * n^2 + 1 * n^1 + 1 * n^0
Since the equation is set in base 10, we have:
1 * n^2 + 1 * n^1 + 1 * n^0 = 37
This is a quadratic equation with n as the variable. We can simplify it:
n^2 + n + 1 = 37
Rearranging the equation:
n^2 + n - 36 = 0
Now, we can solve this quadratic equation for n.
We can either factorize it or use the quadratic formula. Let's use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = 1, and c = -36.
Substituting the values:
n = (-1 ± √(1^2 - 4 * 1 * -36)) / 2 * 1
Simplifying further:
n = (-1 ± √(1 + 144)) / 2
n = (-1 ± √145) / 2
Therefore, the possible values of n in base 10 are (-1 + √145) / 2 and (-1 - √145) / 2.