Determine the value of base x if (211)x (153)8?

Answer 1

In place value notation, each column left, starting from the units column, is worth the base times the column on its right.

In base 10 this gives 1, 10, 10², ...

In base 8 this gives 1, 8, 8², ...

In base x this gives 1, x, x², ...

To convert a number in any base to base 10, multiply each digit by its place value column; this gives:

2x² + x + 1 = 1×8² + 5×8 + 3

→ 2x² + x + 1 = 64 + 40 + 3

→ 2x² + x + 1 = 107

→ 2x² + x - 106 = 0

→ x = (-1 ±√(1² - 4×2×-106))/(2×2)

→ x = (-1 ±√849)/4

→ x = (-1 - √849)/4 or x = (-1 + √849)/4

→ x ≈ -7.5344 or x ≈ 7.0344

A negative base is unlikely, thus the base x is (-1 + √849)/4 (or approx 7.0344)

Answer 2

It is 7.0344, approx. Or more accurately, 7.03440114216673