To solve the expression [tex]\(14 \cdot 2 \left\{30 + 2 \cdot 3 \div \left[4 \div 2 + (7 + 2 + \frac{9}{6})\right]\right\}\)[/tex], we'll follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders (i.e., powers and roots, etc.), Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).
Let's break down this problem step by step:
1. Innermost Expression:
- Start with the innermost division:
[tex]\[
\frac{9}{6} = 1.5
\][/tex]
2. Parentheses Calculation:
- Sum inside the parentheses:
[tex]\[
7 + 2 + 1.5 = 10.5
\][/tex]
3. Division within Brackets:
- Calculate [tex]\(4 \div 2\)[/tex]:
[tex]\[
4 \div 2 = 2
\][/tex]
- Add this result to the sum found earlier inside the parentheses:
[tex]\[
2 + 10.5 = 12.5
\][/tex]
4. Division and Multiplication within Curly Braces:
- Calculate [tex]\(3 \div 12.5\)[/tex]:
[tex]\[
3 \div 12.5 = 0.24
\][/tex]
- Multiply the result by 2:
[tex]\[
0.24 \times 2 = 0.48
\][/tex]
5. Addition inside Curly Braces:
- Add 30 to the result obtained from the previous step:
[tex]\[
30 + 0.48 = 30.48
\][/tex]
6. Outer Multiplication:
- Multiply the above sum by 2:
[tex]\[
2 \times 30.48 = 60.96
\][/tex]
7. Final Multiplication:
- Finally, multiply the whole result by 14:
[tex]\[
14 \times 60.96 = 853.44
\][/tex]
Thus, the final result of the given expression is [tex]\(853.44\)[/tex].
