Let's simplify each of the given expressions step-by-step:
### Expression 1: [tex]\(6 \times n \times 6 \times 6 \times 6 \times n\)[/tex]
1. Group the similar factors together:
[tex]\[
6 \times 6 \times 6 \times 6 \times n \times n
\][/tex]
2. Combine the factors with the same base by adding their exponents:
[tex]\[
6^4 \times n^2
\][/tex]
Simplified form:
[tex]\[
6^4 \times n^2
\][/tex]
### Expression 2: [tex]\(\sqrt{1} \times \sqrt{2} \times \sqrt{50}\)[/tex]
1. Calculate [tex]\(\sqrt{1}\)[/tex]:
[tex]\[
\sqrt{1} = 1
\][/tex]
2. Rewrite [tex]\(\sqrt{50}\)[/tex] in terms of its factors:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \times \sqrt{2}
\][/tex]
3. Substitute these back into the expression:
[tex]\[
1 \times \sqrt{2} \times 5 \times \sqrt{2}
\][/tex]
4. Combine the radicals:
[tex]\[
1 \times 5 \times (\sqrt{2} \times \sqrt{2}) = 5 \times 2 = 10
\][/tex]
Simplified form:
[tex]\[
10
\][/tex]
### Expression 3: [tex]\(\frac{x}{2} \times \frac{1}{2} \times \frac{1}{2}\)[/tex]
1. Rewrite the expression multiplying the fractions together:
[tex]\[
\frac{x}{2} \times \frac{1}{2} \times \frac{1}{2}
\][/tex]
2. Combine the fractions:
[tex]\[
\frac{x \times 1 \times 1}{2 \times 2 \times 2} = \frac{x}{8}
\][/tex]
Simplified form:
[tex]\[
\frac{x}{8}
\][/tex]
### Summary:
1. [tex]\(6^4 \times n^2\)[/tex]
2. [tex]\(10\)[/tex]
3. [tex]\(\frac{x}{8}\)[/tex]
