This problem involves converting numbers from different bases to a common base (base 10) and then solving for an unknown base, x. We are given an equation where a number in base x is equal to a number in base 8. To find the value of base x, we will convert both numbers to base 10, set them equal to each other, and then solve the resulting algebraic equation.

Number Bases Explained

A number base, or radix, defines the number of unique digits, including zero, that a number system uses to represent numbers. For example, the decimal system (base 10) uses ten digits (0-9), and the binary system (base 2) uses two digits (0-1). When converting a number from any base to base 10, we use the following formula:

• If a number is represented as \((d_n d_{n-1} \dots d_1 d_0)_b\), where \(d\) represents a digit and \(b\) is the base, its value in base 10 is calculated as:
• \(d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \dots + d_1 \cdot b^1 + d_0 \cdot b^0\)

Converting Numbers to Base 10

Converting \((211)_x\) to Base 10

Let's convert the number \((211)_x\) from base x to base 10. Here, the digits are 2, 1, and 1. The positions correspond to powers of the base x: \(x^2\), \(x^1\), and \(x^0\).

Using the conversion formula:

\[ (211)_x = 2 \cdot x^2 + 1 \cdot x^1 + 1 \cdot x^0 \]

\[ (211)_x = 2x^2 + x + 1 \]

Converting \((152)_8\) to Base 10

Next, we convert the number \((152)_8\) from base 8 to base 10. The digits are 1, 5, and 2. The positions correspond to powers of the base 8: \(8^2\), \(8^1\), and \(8^0\).

Using the conversion formula:

\[ (152)_8 = 1 \cdot 8^2 + 5 \cdot 8^1 + 2 \cdot 8^0 \]

Let's calculate the values:

• \(1 \cdot 8^2 = 1 \cdot 64 = 64\)
• \(5 \cdot 8^1 = 5 \cdot 8 = 40\)
• \(2 \cdot 8^0 = 2 \cdot 1 = 2\)

Adding these values together gives us the base 10 equivalent:

\[ (152)_8 = 64 + 40 + 2 = 106 \]

Equating the Number Bases

The problem states that the number \((211)_x\) is equal to the number \((152)_8\). Now that we have converted both numbers to their base 10 equivalents, we can set up the equation:

\[ 2x^2 + x + 1 = 106 \]

Solving for Unknown Base x

To find the value of x, we need to solve this quadratic equation. First, rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\):

\[ 2x^2 + x + 1 - 106 = 0 \]

\[ 2x^2 + x - 105 = 0 \]

We can solve this quadratic equation by factorization. We look for two numbers that multiply to \((2 \cdot -105) = -210\) and add up to the coefficient of x, which is 1. The numbers that satisfy these conditions are 15 and -14.

Let's rewrite the middle term (\(x\)) using these two numbers:

\[ 2x^2 + 15x - 14x - 105 = 0 \]

Now, we group the terms and factor out common factors from each group:

\[ x(2x + 15) - 7(2x + 15) = 0 \]

Notice that \((2x + 15)\) is a common factor in both terms. We can factor it out:

\[ (x - 7)(2x + 15) = 0 \]

This equation gives us two possible values for x, by setting each factor to zero:

• Possibility 1: \(x - 7 = 0 \implies x = 7\)
• Possibility 2: \(2x + 15 = 0 \implies 2x = -15 \implies x = -\frac{15}{2}\)

Validating Base Value

For a number base to be considered valid, it must meet two essential conditions:

• The base must always be a positive integer.
• The base must be strictly greater than any individual digit used within the number itself. In the number \((211)_x\), the largest digit present is 2, which means that the base x must be greater than 2 (\(x > 2\)).

Let's check our calculated solutions against these rules:

• For \(x = 7\):
  • 7 is a positive integer.
  • 7 is greater than 2 (the largest digit in \((211)_x\)).
  • Therefore, \(x = 7\) is a valid base for the number system.
• For \(x = -\frac{15}{2}\):
  • \(-\frac{15}{2}\) is not a positive integer (it's a negative fraction).
  • Therefore, \(x = -\frac{15}{2}\) is not a valid base for a number system.

Based on this validation, the only mathematically and practically sound value for base x is 7.

The final answer is 7.