Consider the process of evaluating the numerical expression below using the standard order… — Armed Services Vocational Aptitude Battery (ASVAB)

Consider the process of evaluating the numerical expression below using the standard order of operations:

14 + 3 \times (7 - 2^2) - 8 \div 4

How should the following intermediate simplified forms be arranged to represent the exact chronological sequence of the step-by-step evaluation?

1 $14 + 3 \times (7 - 4) - 8 \div 4$
2 $14 + 3 \times 3 - 8 \div 4$
3 $14 + 9 - 2$
4 $23 - 2$

Answer

The expressions should be ordered starting with the exponent evaluation, followed by the parentheses, then multiplication and division, and finally addition.

The proper application of the order of operations (PEMDAS/BODMAS) ensures that we first simplify inside the parentheses (handling the exponent first, then subtraction), next process multiplication and division from left to right, and finally address addition and subtraction from left to right. This produces the exact sequence of intermediate expressions shown in the correct arrangement.

Step-by-Step Solution

Evaluate the exponent inside the grouping symbols.

2^2

simplifies to

4

, changing the expression to

14 + 3 \times (7 - 4) - 8 \div 4

Order of operations dictates that exponents inside parentheses take precedence over other operations.

Complete the operation inside the parentheses.

7 - 4

simplifies to

3

, yielding

14 + 3 \times 3 - 8 \div 4

All operations within grouping symbols must be fully resolved before proceeding to external operations.

Evaluate multiplication and division.

3 \times 3

becomes

9

and

8 \div 4

becomes

2

, resulting in

14 + 9 - 2

Multiplication and division share the same priority and are executed from left to right before addition or subtraction.

Perform the addition.

14 + 9

becomes

23

, leaving

23 - 2

Addition and subtraction share the lowest priority and are evaluated from left to right.

Perform the final subtraction.

23 - 2

simplifies to the final answer of

21

This is the final remaining mathematical operation.

Key Concept

Order of Operations (PEMDAS)

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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