Problem 97 - Prealgebra

Question

Simplify. $$5 \cdot 2^{4}-3 \cdot 4^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 32.

Step 1: Understanding the Expression

We need to simplify the expression: \[ 5 \cdot 2^{4} - 3 \cdot 4^{2} \].The expression involves both multiplication and exponentiation. Let's handle the exponents before moving to multiplication.

Step 2: Calculate the Exponents

First, compute \(2^4\) and \(4^2\): \[2^4 = 2 \times 2 \times 2 \times 2 = 16\] and \[4^2 = 4 \times 4 = 16\] Now the expression updates to: \[5 \cdot 16 - 3 \cdot 16\]

Step 3: Applying Multiplication

Now, apply the multiplication: Calculate \(5 \cdot 16\): \[5 \times 16 = 80\] Calculate \(3 \cdot 16\): \[3 \times 16 = 48\].The expression now becomes: \[80 - 48\]

Step 4: Performing Subtraction

Subtract 48 from 80 to simplify the expression: \[80 - 48 = 32\].Thus, the simplified expression is 32.

Key Concepts

Understanding Exponents

Exponents represent repeated multiplication of a number by itself. Specifically, when we see an expression like \(2^4\), this means "2 raised to the power of 4," which translates to multiplying 2 by itself four times: \(2 \times 2 \times 2 \times 2 = 16\). Similarly, \(4^2\) means multiplying 4 by itself twice, resulting in \(4 \times 4 = 16\). Exponents are crucial in simplifying expressions because they dictate the number of times a number is multiplied by itself:

• Base: The number being multiplied, for example, the "2" in \(2^4\).
• Exponent: The power to which the base is raised, indicating how many times it is multiplied, like the "4" in \(2^4\).

Recognizing and calculating exponents correctly is the first step in simplifying expressions that include them.

The Role of Multiplication

Multiplication is simply adding a number to itself a certain number of times. When handling expressions with multiplication following exponents, as in our example exercise, we apply multiplication after dealing with the exponents.In the expression \(5 \cdot 2^4 - 3 \cdot 4^2\), once the exponents are calculated, we move on to multiplication:

• Calculate \(5 \times 16\): Here, we multiply 5 with the result of \(2^4\), which is 16. This results in 80.
• Calculate \(3 \times 16\): Similarly, multiplying 3 with 16 from \(4^2\) gives us 48.

Multiplication occurs after exponents but before subtraction. Handling operations in the correct sequence is essential for accurate simplification.

Navigating the Order of Operations

The order of operations is a set of rules that dictate the sequence in which we perform mathematical operations. This ensures expressions are simplified correctly, avoiding any ambiguities. The common acronym PEMDAS helps us remember this order:

• P: Parentheses first
• E: Exponents (ie: powers and roots, etc.)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

In the problem \(5 \cdot 2^4 - 3 \cdot 4^2\), we must apply these rules:- First, handle the exponents: \(2^4\) and \(4^2\).- Then, carry out multiplication, resulting in \(5 \times 16\) and \(3 \times 16\).- Finally, perform the subtraction: \(80 - 48\). This ordered approach ensures clarity and accuracy in mathematical expressions.