Problem 19 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 2^{4}-5 \cdot 3 $$

Step-by-Step Solution

Answer

The evaluated expression is 1.

Step 1: Calculate the Exponentiation

Start off by calculating the value of the exponentiation, \(2^{4}\). This means multiplying 2 by itself four times, which equals 16.

Step 2: Perform the Multiplication

Next, perform the multiplication operation, which is \(5 \cdot 3\). This equals 15.

Step 3: Subtract to get the Answer

Now that you have calculated the result of the exponentiation, which is 16, and the multiplication, which is 15, subtract the multiplication result from the exponentiation result: \(16 - 15 = 1\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation essential for handling powers of numbers. In simpler terms, it's the process of multiplying a number by itself a certain number of times. For this expression, we start with \(2^4\). This means we multiply 2 by itself four times.

• First, multiply 2 by 2 to get 4.
• Then, multiply that 4 by another 2 to get 8.
• And finally, multiply that 8 by the final 2 to reach 16.

Exponentiation is crucial because it forms the initial step in the order of operations, sometimes remembered as "PEMDAS" or "BODMAS," where "E" stands for exponentiation.

By completing this step first, we ensure the correctness of forthcoming operations in solving mathematical expressions.

Multiplication

Multiplication is a fundamental operation in mathematics that simplifies the addition of the same number repeatedly. In this exercise, after exponentiation, the next step is to multiply 5 by 3. The expression \(5 \cdot 3\) can be viewed as adding five, three times:

• First, add 5 to itself to get 10.
• Next, add 5 again to obtain 15.

Multiplication is significant because it follows right after exponentiation in the order of operations. This hierarchy ensures we accurately resolve expressions step-by-step, avoiding potential errors. Ultimately, accurately performing multiplication allows us to confidently proceed to subsequent operations.

Subtraction

Subtraction is the mathematical process of taking one number away from another. It's used to find the difference between two numbers. This operation often follows after dealing with exponentiation and multiplication, as it does in this case. Once we've calculated \(2^4 = 16\) and \(5 \cdot 3 = 15\), we subtract 15 from 16.

• Start with 16 and remove 15.
• This leaves a difference of 1.

Subtraction is crucial in the order of operations because it typically comes near the end. It's important to have correctly sorted through prior calculations like exponentiation and multiplication first. By concluding with subtraction, we arrive at the final, accurate result of the exercise.