Problem 92 - Prealgebra

Question

Use the rule for order of operations to simplify each expression. $$2 \cdot 3^{2}-18 \div 3+2^{4}$$

Step-by-Step Solution

Answer

The simplified expression is 28.

Step 1: Evaluate Exponents

Identify and evaluate the exponents in the expression. Begin with any terms involving exponents. The expression has two terms with exponents: \(3^2\) and \(2^4\). Calculate them first: \(3^2 = 9\) and \(2^4= 16\). The expression now becomes: \(2 \cdot 9 - 18 \div 3 + 16\).

Step 2: Perform Multiplication and Division

Proceed by performing multiplication and division from left to right. First, multiply: \(2 \cdot 9 = 18\).Then, divide: \(18 \div 3 = 6\).The expression now becomes: \(18 - 6 + 16\).

Step 3: Perform Addition and Subtraction

Complete the expression by performing addition and subtraction from left to right. First, subtract: \(18 - 6 = 12\).Then, add: \(12 + 16 = 28\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, the exponent in \(3^2\) means you multiply 3 by itself once: \(3 \times 3 = 9\). Another example is \(2^4\), which means you multiply 2 by itself three more times: \(2 \times 2 \times 2 \times 2 = 16\).
Exponents are dealt with first in the order of operations. This means when you see an equation, calculate any exponential values before moving onto other operations.
Understanding exponents makes larger calculations simpler. Instead of writing \(3 \times 3\), you can simply write \(3^2\). This shorthand is useful when handling complex mathematical expressions, as it ensures clarity and precision in steps.

Multiplication and Division

After handling any exponents in a mathematical expression, you approach multiplication and division next. These two operations are equally important in the order of operations.
One key rule is to move from left to right across the expression, performing any multiplication or division as they appear.
For example, in the portion \(2 \cdot 9 - 18 \div 3 + 16\), you handle the multiplication first: \(2 \cdot 9 = 18\). Then, you perform the division: \(18 \div 3 = 6\).

• Multiplication: \(a \cdot b\) results in the product of \(a\) and \(b\).
• Division: \(a \div b\) means dividing \(a\) by \(b\), yielding a quotient.

This step ensures you have simplified components before moving onto combination with addition and subtraction.

Addition and Subtraction

The final step is handling addition and subtraction, again from left to right. These operations are performed last because they complete the expression, finalizing the result based on previously calculated values.
Using the simplified expression \(18 - 6 + 16\), you first perform the subtraction: \(18 - 6 = 12\). Then, the resulting number is added to the next part: \(12 + 16 = 28\).
Remember:

• Addition involves combining amounts (\(a + b\)).
• Subtraction involves finding the difference between amounts (\(a - b\)).

Executing these operations correctly is crucial as any misstep could change the final result entirely. By maintaining the left-to-right sequence after multiplication and division are complete, clarity and accuracy are ensured.