Evaluating Algebraic Expressions What Is The Value Of Expression ${ }^{9+\frac{n}{3}-6}$ When $n=12$

Introduction

In the realm of mathematics, evaluating expressions is a fundamental skill. It involves substituting given values for variables and simplifying the expression to obtain a numerical result. This article delves into the process of evaluating the algebraic expression ${ }^{9+\frac{n}{3}-6}$ when $n=12$. We will break down the steps involved, providing a clear and concise explanation to enhance understanding. By mastering this skill, you will be well-equipped to tackle more complex mathematical problems.

Understanding the Expression

The expression ${ }^{9+\frac{n}{3}-6}$ is an algebraic expression that involves a variable, $n$. The expression consists of constants (9 and 6), a variable ($n$), and mathematical operations (addition, division, and subtraction). To evaluate this expression, we need to substitute the given value of $n$ into the expression and simplify it using the order of operations (PEMDAS/BODMAS).

Breaking Down the Components

Let's dissect the expression to understand each part:

• Constants: The constants are the numerical values that do not change. In this expression, we have 9 and 6.
• Variable: The variable is a symbol (in this case, $n$) that represents an unknown value. The value of the variable can change.
• Mathematical Operations: The operations in this expression include:
  • Addition (+): Adding numbers together.
  • Division ($\frac{}{}$): Dividing one number by another. Specifically, $\frac{n}{3}$ means $n$ divided by 3.
  • Subtraction (-): Subtracting one number from another.

Order of Operations (PEMDAS/BODMAS)

To simplify the expression correctly, we need to follow the order of operations, often remembered by the acronyms PEMDAS or BODMAS:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our expression, we have division, addition, and subtraction. According to the order of operations, we perform division first, then addition and subtraction from left to right. This structured approach ensures we arrive at the correct answer by addressing each operation in its proper sequence.

Step-by-Step Evaluation

Now, let's evaluate the expression ${ }^{9+\frac{n}{3}-6}$ when $n=12$ step-by-step.

Step 1: Substitute the Value of $n$

The first step is to substitute the given value of $n$, which is 12, into the expression. This means we replace $n$ with 12 in the expression:

${ }^{9+\frac{12}{3}-6}$

Step 2: Perform the Division

Next, we perform the division operation. We have $\frac{12}{3}$, which equals 4. So, the expression becomes:

${ }^{9+4-6}$

Step 3: Perform Addition and Subtraction (from left to right)

Now we have addition and subtraction. We perform these operations from left to right. First, we add 9 and 4:

$9 + 4 = 13$

So, the expression becomes:

${ }^{13-6}$

Step 4: Perform Subtraction

Finally, we subtract 6 from 13:

$13 - 6 = 7$

Therefore, the value of the expression ${ }^{9+\frac{n}{3}-6}$ when $n=12$ is 7. This methodical approach ensures accuracy, breaking down the problem into manageable steps and adhering to the correct order of operations.

Detailed Explanation of Each Step

To further clarify the evaluation process, let's delve into a detailed explanation of each step involved in evaluating the expression ${ }^{9+\frac{n}{3}-6}$ when $n=12$.

Step 1: Substitution

The cornerstone of evaluating algebraic expressions is substitution. When we substitute a value for a variable, we replace the variable with the given numerical value. In our case, we are given that $n=12$. This means that wherever we see the variable $n$ in the expression, we replace it with the number 12. The original expression is ${ }^{9+\frac{n}{3}-6}$. After substituting $n$ with 12, the expression transforms into ${ }^{9+\frac{12}{3}-6}$. This substitution is crucial because it allows us to convert the algebraic expression into a numerical expression, which we can then simplify using arithmetic operations. The process of substitution is not just a mechanical replacement; it’s a fundamental step that bridges the gap between abstract algebra and concrete arithmetic. It’s the initial move in a sequence of operations that ultimately leads to the solution. By accurately substituting the value, we set the stage for the subsequent steps, ensuring that our calculations are based on the correct numerical inputs. This meticulous attention to detail at the outset is paramount for achieving the correct final result. The substitution step is a clear and direct application of a given condition, turning an expression with a variable into a calculable form.

Step 2: Division

Following the order of operations, our next focus is on division. In the expression ${ }^{9+\frac{12}{3}-6}$, the division operation is represented by the fraction $\frac{12}{3}$. According to the order of operations, division takes precedence over addition and subtraction. Therefore, before we can proceed with any addition or subtraction, we must simplify this fraction. The fraction $\frac{12}{3}$ means