Certainly! Let's break down and solve each mathematical expression step by step.
### Question 6: [tex]\(\{5(21 \div \overline{7-4})-30\}+2 \times 10 \div 5\)[/tex]
1. Start with the innermost parentheses:
[tex]\[
7 - 4 = 3
\][/tex]
2. Substitute [tex]\(7 - 4\)[/tex] with 3:
[tex]\[
5(21 \div 3) - 30 + 2 \times 10 \div 5
\][/tex]
3. Perform the division inside the parentheses next:
[tex]\[
21 \div 3 = 7
\][/tex]
4. Substitute [tex]\(21 \div 3\)[/tex] with 7:
[tex]\[
5 \times 7 - 30 + 2 \times 10 \div 5
\][/tex]
5. Perform the multiplication:
[tex]\[
5 \times 7 = 35
\][/tex]
6. Substitute [tex]\(5 \times 7\)[/tex] with 35:
[tex]\[
35 - 30 + 2 \times 10 \div 5
\][/tex]
7. Perform the multiplication and the division from left to right:
[tex]\[
2 \times 10 = 20
\][/tex]
[tex]\[
20 \div 5 = 4
\][/tex]
8. Substitute fully:
[tex]\[
35 - 30 + 4
\][/tex]
9. Perform the subtraction and addition sequentially:
[tex]\[
35 - 30 = 5
\][/tex]
[tex]\[
5 + 4 = 9
\][/tex]
Therefore, the answer to Question 6 is:
[tex]\[
\boxed{9}
\][/tex]
### Question 7: [tex]\(1-[1-\{1-\overline{1-1})\}]\)[/tex]
1. Start with the innermost operation:
[tex]\[
1 - 1 = 0
\][/tex]
2. Substitute [tex]\(1 - 1\)[/tex] with 0:
[tex]\[
1 - [1 - \{1 - 0\}]
\][/tex]
3. Now simplify inside the curly brackets:
[tex]\[
1 - 0 = 1
\][/tex]
4. Substitute [tex]\(1 - 0\)[/tex] with 1:
[tex]\[
1 - [1 - 1]
\][/tex]
5. Simplify inside the brackets:
[tex]\[
1 - 1 = 0
\][/tex]
6. Substitute [tex]\(1 - 1\)[/tex] with 0:
[tex]\[
1 - 0
\][/tex]
7. Simplify the final operation:
[tex]\[
1 - 0 = 1
\][/tex]
Therefore, the answer to Question 7 is:
[tex]\[
\boxed{1}
\][/tex]
