Problem 34 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when a = 3 and c = 5. $$ a \cdot\left(c^{2}\right) $$

Step-by-Step Solution

Answer

The evaluated expression is 75

Step 1: Substitute the Given Values

Replace the variables a and c in the expression with the provided values. So, the expression becomes \(3 \cdot (5^{2})\).

Step 2: Resolve the Exponent

Simplify inside the parenthesis by squaring the number 5 which equals to 25.

Step 3: Multiply

Multiply 3 with 25 to get the final result.

Key Concepts

Substituting Values

To evaluate variable expressions, one of the first steps is often substituting values. This process involves replacing the variables in an algebraic expression with actual numbers. Doing so simplifies the task from an abstract one to a more concrete calculation. Think of variables as empty boxes waiting to be filled – once we know the content, we can proceed to use it in our formula.

For instance, if you have an algebraic formula where the variable a appears, and you're given that a = 3, you simply replace every occurrence of a in the expression with the number 3. Similarly, if c were another variable with a value of 5, every c in your expression would become 5. This step is crucial because it sets the stage for the actual arithmetic that follows.

Keep in mind that careful substitution is important to avoid any mix-up of values, which could lead to incorrect results. Always double-check that you have substituted the correct values for each variable before moving on to the next operation.

Resolving Exponents

Once the values have been substituted, we often encounter resolving exponents as the next step. An exponent tells us how many times to use the number in a multiplication. It's written as a small number to the top right of the base number. For instance, if you see 5², it means 5 multiplied by itself, which is 5 × 5 = 25. This is known as 'squaring' because the exponent is 2.

Resolving exponents is fundamental to simplifying expressions because it affects the magnitude of numbers greatly. For example, raising a number to the third power (cubing it) would be even bigger than squaring.

It's also important to remember that exponents are dealt with before multiplication or division operations, in line with the order of operations. If you encounter a negative base with an exponent, remember to consider whether the exponent is even (which would result in a positive outcome) or odd (resulting in a negative outcome).

Multiplication in Algebra

Moving ahead in the process of evaluating expressions, once we've substituted values and resolved any exponents, we often come across multiplication in algebra. Multiplying numbers in algebra is similar to simple arithmetic, but it might involve numbers, variables, or a combination of both.

In our example, after resolving the exponent, we are left with the multiplication of two numbers. Just like basic multiplication, we multiply these numbers together. So 3 × 25 would give us 75. Note that if we had variables instead of numbers, we would multiply the coefficients (the numbers in front of the variables) and apply the laws of exponents for any variables involved. Proper multiplication is critical for obtaining a correct answer in any algebraic expression.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is a hierarchical framework for tackling math problems that involve more than one operation.

Following this order is essential because performing operations out of order can lead to completely different results. The reason for this hierarchy is to have a universal standard so that everyone can arrive at the same answer for the same mathematical expression. First, deal with what's inside the parentheses or grouping symbols, then move to exponents. After that, carry out any multiplication or division from left to right, and finally, perform any addition or subtraction, also from left to right.

The order of operations must be strictly observed for accurate solutions, particularly in complex expressions that include several different operations. It's like following a recipe; the steps taken in the correct sequence ensure the desired outcome.