Sure, let's solve each part of the question step-by-step.
### Part 1: [tex]\(\left(\frac{-6}{7}\right) \div \left(\frac{-41}{42}\right)\)[/tex]
1. Understand the division of fractions: Dividing by a fraction is equivalent to multiplying by its reciprocal. So, we need to find the reciprocal of [tex]\(\frac{\text{-41}}{\text{42}}\)[/tex], which is [tex]\(\frac{\text{42}}{\text{-41}}\)[/tex].
2. Rewrite the problem: [tex]\(\left(\frac{-6}{7}\right) \div \left(\frac{-41}{42}\right)\)[/tex] becomes [tex]\(\left(\frac{-6}{7}\right) \times \left(\frac{42}{-41}\right)\)[/tex].
3. Multiply the numerators and the denominators:
[tex]\[
\frac{-6 \cdot 42}{7 \cdot -41} = \frac{-252}{-287}
\][/tex]
4. Simplify the fraction: Both [tex]\(-252\)[/tex] and [tex]\(-287\)[/tex] are negative, so the fraction simplifies to a positive value. They have no common factors to further simplify, so:
[tex]\[
\frac{252}{287} \approx 0.8780487804878049
\][/tex]
So, [tex]\(\left(\frac{-6}{7}\right) \div \left(\frac{-41}{42}\right) = 0.8780487804878049\)[/tex].
### Part 2: [tex]\(\left(-\frac{8}{9}\right) \div \frac{71}{72}\)[/tex]
1. Understand the division of fractions: As before, dividing by a fraction is equivalent to multiplying by its reciprocal. We need the reciprocal of [tex]\(\frac{71}{72}\)[/tex], which is [tex]\(\frac{72}{71}\)[/tex].
2. Rewrite the problem: [tex]\(\left(-\frac{8}{9}\right) \div \frac{71}{72}\)[/tex] becomes [tex]\(\left(-\frac{8}{9}\right) \times \left(\frac{72}{71}\right)\)[/tex].
3. Multiply the numerators and the denominators:
[tex]\[
\frac{-8 \cdot 72}{9 \cdot 71} = \frac{-576}{639}
\][/tex]
4. Simplify the fraction: \\
Both [tex]\(-576\)[/tex] and [tex]\(639\)[/tex] can be divided by their greatest common divisor (which is not needed in exact calculation), but the result is already in its simplest form and negative since only one sign is negative:
[tex]\[
\frac{-576}{639} \approx -0.9014084507042253
\][/tex]
So, [tex]\(\left(-\frac{8}{9}\right) \div \frac{71}{72} = -0.9014084507042253\)[/tex].
### Summary of Results
- [tex]\(\left(\frac{-6}{7}\right) \div \left(\frac{-41}{42}\right) = 0.8780487804878049 \)[/tex]
- [tex]\(\left(-\frac{8}{9}\right) \div \frac{71}{72} = -0.9014084507042253 \)[/tex]
These are the final results for the given fractions.
