Question - Finding the Base of a Number
Solution:
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 = -15Since the base of a number cannot be negative, we disregard r = -15.Therefore, the value of r is 8.
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