- (4) \(\boxed{126}\)
- (5) \(\boxed{86}\)
- (6) \(\boxed{18}\)
- (7) \(\boxed{196}\)
- (8) \(\boxed{2}\)
- (9) \(\boxed{27}\)
- (10) \(\boxed{99}\)
- (11) \(\boxed{4102}\)
- (12) \(\boxed{109}\)
Explanation
- Analysis of the Problem
We are tasked with evaluating nine different mathematical expressions labeled (4) through (12) by applying the order of operations, which dictates that operations in parentheses are performed first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Additionally, for expressions (11) and (12), we use the provided variable values \(x = 8\), \(m = 4\), and \(n = 5\).
- Evaluating Expressions (4) through (6)
(4) \(2(4^3 - 1)\):
First, evaluate the exponent: \(4^3 = 64\).
Subtract inside the parentheses: \(64 - 1 = 63\).
Multiply by 2: \(2 \times 63 = 126\).
(5) \(5 + 9(1 + 2)^2\):
Add inside the parentheses: \(1 + 2 = 3\).
Square the result: \(3^2 = 9\).
Multiply by 9: \(9 \times 9 = 81\).
Add 5: \(5 + 81 = 86\).
(6) \(5 + 9(1) + 2^2\):
Evaluate the exponent: \(2^2 = 4\).
Perform the multiplication: \(9 \times 1 = 9\).
Add the terms: \(5 + 9 + 4 = 18\).
- Evaluating Expressions (7) through (9)
(7) \((18 - 4)^2\):
Subtract inside the parentheses: \(18 - 4 = 14\).
Square the result: \(14^2 = 196\).
(8) \(18 - 4^2\):
Evaluate the exponent: \(4^2 = 16\).
Subtract: \(18 - 16 = 2\).
(9) \(9 + 2(3^2)\):
Evaluate the exponent: \(3^2 = 9\).
Multiply by 2: \(2 \times 9 = 18\).
Add: \(9 + 18 = 27\).
- Evaluating Expressions (10) through (12) and Finalizing Calculations
(10) \((9 + 2)3^2\):
Add inside the parentheses: \(9 + 2 = 11\).
Evaluate the exponent: \(3^2 = 9\).
Multiply the results: \(11 \times 9 = 99\).
(11) \(12 + x^4 - 6\) when \(x = 8\):
Substitute \(x = 8\): \(12 + 8^4 - 6\).
Evaluate the exponent: \(8^4 = 4096\).
Add and subtract: \(12 + 4096 - 6 = 4108 - 6 = 4102\).
(12) \(m^3 + 9n\) when \(m = 4\) and \(n = 5\):
Substitute values: \(4^3 + 9 \times 5\).
Calculate power and product: \(64 + 45 = 109\).
Examples
Understanding the order of operations is crucial in everyday tasks like programming or financial planning. For instance, when calculating a total budget, if you have a base cost, a tax rate applied to a subtotal, and a discount, following the correct order (parentheses, exponents, multiplication, addition) ensures that your budget is accurate. Just like in these expressions, knowing whether to add before multiplying or vice versa prevents costly errors in project management or personal finance.