Problem 31 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 16+8 \cdot 2^{2} $$

Step-by-Step Solution

Answer

The result of evaluating the expression \(16 + 8 \cdot 2^{2}\) is 48.

Step 1: Solve the Exponent

The exponent operation is solved first. Here, \(2^{2}\) equals 4.

Step 2: Perform the Multiplication

After solving the exponent, the multiplication operation is performed next. Multiply 8 by the answer from step 1. So, \(8 \cdot 4\) equals 32.

Step 3: Perform the Addition

Lastly, add the result from step 2 to 16. So, \(16 + 32\) equals 48.

Key Concepts

Exponents

Exponents are a fundamental part of algebra and mathematics in general. They represent repeated multiplication of a number by itself. For example, when we write \(2^{2}\), this means that 2 is multiplied by itself. So, \(2^{2} = 2 \times 2 = 4\). Exponents are sometimes called powers and can also be viewed as a means to compactly display numbers. When solving expressions involving exponents, it is crucial to handle them first because of the rules defined by the Order of Operations.
This ensures expressions are evaluated correctly and consistently across different problems.

Multiplication

Multiplication is a basic arithmetic operation where one number is scaled by another. In the context of order of operations, multiplication comes after exponents in terms of calculating expressions. In our exercise, we multiply 8 by the result of \(2^{2}\) which we found to be 4.
Therefore, the multiplication involves calculating \(8 \times 4\). The result here is 32.
Multiplication is also often visualized as repeated addition. For example, \(8 \times 4\) can be thought of as 8 added together 4 times. This emphasizes how multiplication scales numbers and can solve problems involving groups of items or repeated actions.

Addition

Addition is one of the simplest mathematical operations, often the first arithmetic concept learned. It involves finding the total of different numbers. In our example, after calculating the exponent and multiplication, we add the result of the multiplication to another number.
This means taking 16 and adding it to 32 (the product of 8 and 4 from the earlier steps). When we add these, we get \(16 + 32 = 48\).
Addition allows us to combine values and is frequently used in conjunction with other operations to simplify results or provide final totals.