Finding the Base of a Number
To solve for r when 175_r = 125, we must assume that r stands for the base of the number 175, and we are trying to find a base such that 175 in that base equals 125 in base 10.
Firstly, let's express 125 in base 10 as a sum of powers of the base r:
125 in base 10 can be broken down as:
\(1 \times r^2 + 7 \times r^1 + 5 \times r^0\)
Now, we need to solve for r:
\(1r^2 + 7r + 5 = 125\)
Moving all the terms to one side we get:
\(r^2 + 7r + 5 - 125 = 0\)
Simplify the equation:
\(r^2 + 7r - 120 = 0\)
Now factor the quadratic equation:
\((r - 8)(r + 15) = 0\)
This gives us two possible values for r:
r = 8 or r = -15
Since the base of a number cannot be negative, we disregard r = -15.
Therefore, the value of r is 8.
