Determine the value of base x if (211)x = (6A)16

To find the value of the base x given that (211)_x = (6A)_16, we convert both numbers to their decimal equivalents and then solve for x.

First, let's convert the hexadecimal number (6A)_16 to its decimal equivalent. In the hexadecimal base (base 16), the digits 0-9 have the same value as in decimal and the letters A-F represent the values 10-15, respectively. Therefore, for the number (6A)_16:

The 'A' represents the decimal value 10. The '6' represents the decimal value 6.

Converting (6A)_16 to decimal, we get: (6 * 16^1) + (10 * 16^0) = (6 * 16) + (10 * 1) = 96 + 10 = 106

Now we know that (211)_x is also equal to 106 in decimal.

Next, let’s convert (211)_x to its decimal equivalent: (2 * x^2) + (1 * x^1) + (1 * x^0) = 2x^2 + x + 1

Since this is equal to 106, we have the equation: 2x^2 + x + 1 = 106

Solving for x by rearranging the equation: 2x^2 + x + 1 - 106 = 0 2x^2 + x - 105 = 0

This quadratic equation can be factored (if possible) or solved using the quadratic formula. Let's see if we can factor it first: (2x + 15)(x - 7) = 0

Setting each factor equal to zero, we get two possible solutions for x: 2x + 15 = 0 --> x = -15/2 (Not a valid base, because a base cannot be negative or a fraction) x - 7 = 0 --> x = 7 (A valid base)

So, the value of x that satisfies the equation (211)_x = (6A)_16 is x = 7.