To solve the problem of determining the value of [tex]\( x \)[/tex] for which [tex]\( 32_{\text{four}} = 22_x \)[/tex], we need to follow these steps:
### Step 1: Convert [tex]\( 32_{\text{four}} \)[/tex] to Base 10
The number [tex]\( 32_{\text{four}} \)[/tex] is in base 4. To convert it to base 10, we use the positional value of each digit.
Given [tex]\( 32_{\text{four}} \)[/tex]:
[tex]\[ 32_{\text{four}} = 3 \times 4^1 + 2 \times 4^0 \][/tex]
Calculate each term:
[tex]\[ 3 \times 4^1 = 3 \times 4 = 12 \][/tex]
[tex]\[ 2 \times 4^0 = 2 \times 1 = 2 \][/tex]
Sum the results:
[tex]\[ 12 + 2 = 14 \][/tex]
So, [tex]\( 32_{\text{four}} \)[/tex] in base 10 is 14.
### Step 2: Express [tex]\( 22_x \)[/tex] in Base 10
We need to determine the base [tex]\( x \)[/tex] such that [tex]\( 22_x \)[/tex] equals 14 in base 10. The number [tex]\( 22_x \)[/tex] can be represented as:
[tex]\[ 22_x = 2 \times x^1 + 2 \times x^0 \][/tex]
Simplify this:
[tex]\[ 22_x = 2x + 2 \][/tex]
We know from Step 1 that this expression must equal 14 (since [tex]\( 32_{\text{four}} = 14 \)[/tex]):
[tex]\[ 2x + 2 = 14 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[ 2x + 2 = 14 \][/tex]
Subtract 2 from both sides:
[tex]\[ 2x = 12 \][/tex]
Divide by 2:
[tex]\[ x = 6 \][/tex]
### Conclusion
The base [tex]\( x \)[/tex] for which [tex]\( 32_{\text{four}} = 22_x \)[/tex] is:
[tex]\[ x = 6 \][/tex]
Therefore, the correct answer is [tex]\( \mathbf{six} \)[/tex].
