Problem 88 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$12 d^{2} \text { when } d=9$$

Step-by-Step Solution

Answer

The evaluated expression when \(d = 9\) in \(12d^2\) is 972.

Step 1: Substituting The Value of The Variable

We are supposed to replace the variable 'd' in the given expression \(12d^2\) with the value given, which is 9. So, substitute \(d\) with \(9\) in the expression. This transforms the expression to: \(12(9)^2\).

Step 2: Simplify The Expression

Next, simplify the expression. Remember the order of operations BIDMAS (brackets, indices, division and multiplication, addition and subtraction). The exponent operation takes precedence over multiplication, so we must square 9 before proceeding with the multiplication operation. Hence, the expression becomes: \(12 \times 81\). Multiplying 12 and 81 together, we ultimately get \(972\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, it's crucial to follow the order of operations to ensure the correct outcome. The standard rule to remember is BIDMAS:
- **B**rackets first
- **I**ndices (exponents) next
- **D**ivision and **M**ultiplication (from left to right)
- **A**ddition and **S**ubtraction (from left to right)In our example, the expression is transformed to \(12(9)^2\) after substituting \(d = 9\). According to BIDMAS, we handle the exponentiation first by squaring 9, leading to 81. Once the exponentiation is complete, we perform the multiplication operation, resulting in the final answer of 972. By following BIDMAS, you ensure each part of the expression is evaluated in the correct order, preventing errors and miscalculations.

Substitution Method

The substitution method involves replacing a variable in an expression with a specific given value. This is particularly helpful when evaluating expressions at a particular point. In the exercise, we were given the expression \(12d^2\) and asked to evaluate it for \(d = 9\). To apply the substitution method, follow these steps:
- Identify the variable in the expression (in this case, \(d\)).
- Replace each occurrence of the variable with its given value (here, \(9\)).
- Simplify the expression using the order of operations.After substituting \(d\) with \(9\), the expression became \(12(9)^2\). This substitution is the first step in transforming the expression into a more manageable form for evaluation.

Exponents

Exponents refer to the operation of raising a number to a certain power, which means multiplying that number by itself a specific number of times. In our exercise, the expression \(12d^2\) incorporates the exponent \(^2\), indicating that the variable \(d\) should be squared.Here's how you handle exponents in such expressions:
- Identify the base number, which is the number being raised to the power (here it's \(9\) after substitution).
- Determine the exponent, which tells you how many times the base is multiplied by itself (in this case, 2).
For \(9^2\), you multiply 9 by itself: \(9 \times 9\), resulting in 81. Exponents simplify expressions by providing a quick way to deal with repeated multiplication, making complex calculations more manageable.