Problem 72 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \((4 \cdot 6)^{2}.\) A) 48 B) 96 C) 144 D) 576

Step-by-Step Solution

Answer

D) 576

Step 1: Examine the expression

The given expression is \((4 \cdot 6)^{2}\). According to BIDMAS/BODMAS/PEDMAS rule, the operations inside parentheses must be performed first.

Step 2: Perform multiplication

Multiply the numbers inside the parentheses first: \(4 \cdot 6 = 24\). So the expression now becomes \(24^{2}\).

Step 3: Apply the exponent

Now complete the exponentiation, \(24^{2} = 576\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It determines which procedures to perform first to arrive at the correct answer. A common acronym to remember the sequence is BIDMAS, BODMAS, or PEMDAS, depending on where you're from. This stands for Brackets (or Parentheses), Indices (or Exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

Let's apply this to the exercise \( (4 \cdot 6)^{2} \). The parentheses indicate the multiplication must be done first, followed by the exponentiation. The order is important; if we were to square the numbers before multiplying, we would not get the same result. When solving this type of problem:

• First, resolve any calculations inside the parentheses.
• Next, apply the exponent to the result of the first step.
• Continue with multiplication or division as they appear from left to right.
• Finally, tackle any addition or subtraction, also from left to right.

By carefully following these steps, we can avoid mistakes and ensure that we're solving algebraic expressions correctly.

Exponentiation in Algebra

Exponentiation is an operation involving two numbers, the base and the exponent. In algebra, when a number is raised to an exponent, it means you multiply the base by itself as many times as the exponent indicates. For instance, \(24^{2}\) is read as '24 squared', meaning \(24\) multiplied by itself: \(24 \times 24\).

Exponents can be more than just squares: \(a^{n}\) represents 'a' raised to the power of 'n', hinting at multiplicating 'a' by itself 'n' times. Here are some key points about exponentiation:

• A positive exponent indicates the number of times the base is multiplied by itself.
• If the exponent is zero (\(a^{0}\)), the result is always 1, assuming 'a' is not zero.
• A negative exponent \(a^{-n}\) represents the reciprocal of the base raised to the positive exponent \(1/a^{n}\).

Correctly applying exponentiation in our exercise \( (4 \cdot 6)^{2} \) meant interpreting \(24^{2}\) as \(24 \times 24\), which leads to the correct answer of 576.

Solving Algebraic Expressions

To solve an algebraic expression means to simplify it or find its value given certain numbers. It's a series of steps that must be followed in order; knowing the order of operations and how to apply exponentiation is vital. Algebraic expressions can include variables and constants, operations such as addition, subtraction, multiplication, division, and exponents.

When solving, like in our exercise \( (4 \cdot 6)^{2} \), there are no variables involved, so the steps simplifies to:

• First evaluate the expression inside the parentheses.
• Then apply exponents to the result as necessary.

For expressions with variables, you would perform similar steps, substituting the variable with its value when known. It's essential to work methodically and keep the order of operations in mind to avoid errors. As expressions become more complex, these fundamental skills will be the foundation for correctly evaluating them and arriving at the solution.