Problem 54 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(4^{2}-10 \div 2.\) $$A)3\quad B)11\quad C)13\quad D)21$$

Step-by-Step Solution

Answer

B) 11

Step 1: Perform Exponentiation

The first operation to perform is the exponentiation. This is to say, \(4^{2}\) will be calculated first which equals 16.

Step 2: Perform Division

The next operation to perform is the division. This is, \(10 \div 2\) which equals 5.

Step 3: Perform Subtraction

The last operation to perform is the subtraction. This is, subtract the result from the division (5), from the result of the exponentiation (16). This is \(16-5\), which equals 11.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating expressions in algebra. It provides a clear protocol to follow, ensuring that everyone solves mathematical problems consistently and correctly. The acronym PEMDAS can help us remember this order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Let's apply this rule to the given expression, which is to evaluate \(4^{2}-10 \div 2\). Here, no parentheses dictate a different order, so we look for exponents first. That's why we calculate \(4^{2}\) in the first step, as per the order of operations. Next, division comes before subtraction, so the division of 10 by 2 is carried out. Only after these steps is it appropriate to perform the subtraction.

Exponentiation

In simplifying algebraic expressions, exponentiation is a fundamental operation. It represents repeated multiplication of the same number. The expression \(a^{n}\) implies that 'a' is multiplied by itself 'n' times. For example, \(4^{2}\) signifies 4 multiplied by itself, which equals 16.

When dealing with exponentiation, it's crucial to remember that it takes precedence over most other operations, as stated in the order of operations. So when you come across an expression with an exponent, such as the one in our exercise, perform this calculation first before moving on to subsequent steps. By getting a firm grasp on how exponentiation works, you can solve more complex equations with confidence.

Arithmetic Operations

The bedrock of algebraic expressions includes various arithmetic operations: addition, subtraction, multiplication, and division. These actions are the backbone of not just algebra, but all of mathematics.

After exponentiation, we proceed with multiplication or division, whichever comes first from left to right, followed by addition or subtraction in the same manner. In the exercise at hand, after exponentiation, we had a division operation (\(10 \div 2\)), which simplifies to 5. As for the last operation, subtraction, we take the result of exponentiation and subtract the result of division, yielding 11, as shown in the problem's step-by-step solution. By mastering these operations and their hierarchy, students can tackle a wide variety of problems.