Problem 12 - Beginning and Intermediate Algebra

Question

For each expression, label the order in which the operations should be performed. Do not actually perform them. $$ 9-2^{3}+3 \cdot 4 $$

Step-by-Step Solution

Answer

1. Exponent \(2^3\), 2. Multiplication \(3 \times 4\), 3. Subtraction and addition (left to right) \(9 - 8 + 12\).

Step 1: Identify the Exponents

First, identify and label any exponents in the expression. Exponents take precedence over addition, subtraction, and multiplication.Expression: \[9 - 2^3 + 3 \times 4\]Identify: \[2^3\]

Step 2: Identify Multiplication and Division

Next, look for any multiplication or division operations. These operations are performed after exponents but before addition and subtraction.Expression: \[9 - 8 + 3 \times 4\]Identify: \[3 \times 4\]

Step 3: Identify Addition and Subtraction

Finally, locate any addition and subtraction operations. These operations are performed last, from left to right.Expression: \[1 + 12\ (i.e., the result after performing previous steps)\]

Key Concepts

exponents

Understanding exponents is crucial in the order of operations. An exponent tells you how many times to multiply a number by itself. For example, in the expression \(2^3\), the number 2, called the base, is multiplied by itself three times: \(2 \times 2 \times 2 = 8\). Exponents are always performed first in an expression, before any multiplication, division, addition, or subtraction. Let’s look at our example expression: \( 9 - 2^3 + 3 \times 4\). First, we need to handle the exponent. Here, \(2^3 = 8\). After simplifying the exponent, the expression becomes: \(9 - 8 + 3 \times 4\).

multiplication and division

After dealing with exponents, the next step is to perform any multiplication or division operations. These operations are of equal importance and are handled from left to right as they appear in the expression. In our example expression, \(9 - 8 + 3 \times 4\), we need to perform the multiplication next. When you see \(3 \times 4\), you multiply the numbers to get \(12\). Now the expression looks like this: \(9 - 8 + 12\). Multiplication and division must always be completed before moving on to addition and subtraction.

addition and subtraction

The final step in the order of operations involves handling addition and subtraction. These operations are also of equal precedence and should be performed from left to right as they appear in the expression. Continuing with our simplified example, we have: \(9 - 8 + 12\). First, perform the subtraction: \(9 - 8 = 1\). Then, add the remaining number: \(1 + 12 = 13\).Remember to always go from left to right when performing addition and subtraction. By following the correct order of operations—exponents first, followed by multiplication and division, and finally addition and subtraction—you ensure that you get the correct answer.