To multiply the given expressions, let's examine each fraction step by step and identify any issues that arise.
### Step 1: Evaluate each fraction
1. First Term:
[tex]\[
\frac{1}{1+1}
\][/tex]
- Simplify the denominator:
[tex]\[
1 + 1 = 2
\][/tex]
- The first term simplifies to:
[tex]\[
\frac{1}{2} = 0.5
\][/tex]
2. Second Term:
[tex]\[
\frac{1-1}{1-1}
\][/tex]
- Simplify the numerator:
[tex]\[
1 - 1 = 0
\][/tex]
- Simplify the denominator:
[tex]\[
1 - 1 = 0
\][/tex]
- Since the denominator becomes zero, the fraction [tex]\(\frac{0}{0}\)[/tex] is undefined.
### Step 2: Multiply the fractions
Next, we attempt to multiply the simplified fractions:
[tex]\[
0.5 \cdot \frac{0}{0}
\][/tex]
Since the second term is undefined ([tex]\(\frac{0}{0}\)[/tex]), multiplying by it also results in an undefined product.
### Step 3: Conclusion
So, we have:
- The first term simplifies to [tex]\(0.5\)[/tex].
- The second term is undefined.
- The product of [tex]\(0.5\)[/tex] and an undefined term is also undefined.
### Algebraic Relationships
We also need to understand how the simplified terms relate to each other and to the complex number [tex]\((1+1)\)[/tex]:
1. First Term Relation:
- The first term, [tex]\(\frac{1}{1+1}\)[/tex], directly simplifies to [tex]\(\frac{1}{2}\)[/tex], showing a direct relationship to the sum within the denominator.
2. Second Term Relation:
- The second term, [tex]\(\frac{1-1}{1-1}\)[/tex], has both the numerator and denominator simplify to zero, leading to an undefined term, indicating that there is no valid relationship for this fraction as a division by zero is undefined.
### Final Result
In summary, the detailed solution reveals that:
[tex]\[
\frac{1}{1+1} \cdot \frac{1-1}{1-1} = 0.5 \cdot \text{undefined} = \text{undefined}
\][/tex]
Therefore, the product of these terms is indeed undefined. The first term simplifies to 0.5, while the second term, and thus their product, is undefined.
