Evaluate The Expression 18 ÷ 6 × M ÷ 3^n For M=18 And N=2

In the realm of mathematics, evaluating expressions is a fundamental skill. This article will guide you through the process of finding the value of the expression $18 \div 6 \times m \div 3^n$ when $m = 18$ and $n = 2$. This exercise not only reinforces the order of operations but also highlights the importance of careful substitution and calculation. By understanding these principles, you'll be well-equipped to tackle more complex mathematical problems.

Understanding the Order of Operations

Before we dive into the specific expression, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. Let's break down what each letter represents:

• Parentheses: Operations within parentheses (or brackets) are performed first.
• Exponents: Exponents (powers) are evaluated next.
• Multiplication and Division: These operations are performed from left to right.
• Addition and Subtraction: These operations are performed last, also from left to right.

By adhering to PEMDAS, we ensure consistency and accuracy in our calculations. Ignoring this order can lead to incorrect results, emphasizing its significance in mathematical problem-solving.

Applying PEMDAS to Our Expression

Now, let's apply the order of operations to our expression: $18 \div 6 \times m \div 3^n$. Following PEMDAS, we first address any parentheses. In this case, there are none. Next, we look for exponents. We have $3^n$, where $n = 2$. So, we'll evaluate this exponential term first. After exponents, we handle multiplication and division from left to right. Finally, we would address addition and subtraction, but these operations are not present in our expression.

Understanding and consistently applying the order of operations is paramount to successfully evaluating mathematical expressions. Let's move on to substituting the given values and simplifying the expression step-by-step.

Substituting the Values of m and n

The next crucial step in evaluating our expression is to substitute the given values for the variables. We are given that $m = 18$ and $n = 2$. Replacing these variables in the expression $18 \div 6 \times m \div 3^n$, we get:

$18 \div 6 \times 18 \div 3^2$

This substitution transforms the algebraic expression into an arithmetic one, which we can now simplify using the order of operations. The act of substitution is a fundamental skill in algebra, allowing us to move from general expressions to specific numerical values. It's essential to ensure accuracy during this step, as any error in substitution will propagate through the rest of the calculation.

The Importance of Accurate Substitution

Accurate substitution is the cornerstone of solving algebraic problems. A simple mistake in replacing a variable with its value can lead to a completely wrong answer. Therefore, double-checking your substitutions is always a good practice. This meticulous approach not only ensures accuracy but also builds confidence in your problem-solving abilities. In our case, we've carefully replaced m with 18 and n with 2, setting the stage for the next phase of simplification.

Now that we've successfully substituted the values, the expression is ready for simplification using the order of operations. Let's proceed with evaluating the exponential term first, as dictated by PEMDAS.

Evaluating the Exponential Term

Following the order of operations, the next step after substitution is to evaluate the exponential term. In our expression, $18 \div 6 \times 18 \div 3^2$, the exponential term is $3^2$. This means 3 raised to the power of 2, which is $3 \times 3$. Calculating this, we get:

$3^2 = 3 \times 3 = 9$

Now, we can replace $3^2$ with 9 in our expression, which becomes:

$18 \div 6 \times 18 \div 9$

Evaluating exponents is a key part of simplifying expressions, and understanding how exponents work is crucial for further mathematical studies. In this case, we've correctly evaluated $3^2$, which allows us to continue simplifying the expression.

Exponents: A Quick Review

To reinforce our understanding, let's briefly review exponents. An exponent indicates how many times a base number is multiplied by itself. For example, in $3^2$, 3 is the base and 2 is the exponent. This means we multiply 3 by itself 2 times. Similarly, $2^3$ would mean $2 \times 2 \times 2 = 8$. A solid grasp of exponents is essential for various mathematical concepts, from polynomial expressions to scientific notation.

With the exponential term evaluated, we're now ready to tackle the remaining operations in our expression. According to PEMDAS, we'll handle multiplication and division from left to right in the next section.

Performing Multiplication and Division

With the exponential term simplified, our expression now reads $18 \div 6 \times 18 \div 9$. According to the order of operations (PEMDAS), we perform multiplication and division from left to right. This means we'll start with the leftmost operation, which is division in this case.

Step-by-Step Calculation

1. First Division: $18 \div 6 = 3$. Our expression now becomes $3 \times 18 \div 9$.
2. Multiplication: Next, we perform the multiplication: $3 \times 18 = 54$. The expression is now $54 \div 9$.
3. Second Division: Finally, we perform the last division: $54 \div 9 = 6$.

Therefore, the value of the expression is 6. By meticulously following the order of operations, we have successfully simplified the expression and arrived at the correct answer. This step-by-step approach minimizes the risk of errors and ensures clarity in our calculations.

Multiplication and Division: Equal Priority

It's important to remember that multiplication and division have equal priority in the order of operations. This is why we perform these operations from left to right. Similarly, addition and subtraction also have equal priority and are performed from left to right. Understanding this equal priority is crucial for accurately simplifying expressions.

Final Result and Conclusion

After carefully following the order of operations and performing each calculation step-by-step, we have arrived at the final result for the expression $18 \div 6 \times m \div 3^n$ when $m = 18$ and $n = 2$. The final value of the expression is:

$6$

This result demonstrates the power of PEMDAS and the importance of methodical problem-solving in mathematics. By understanding and applying the order of operations, we can confidently tackle complex expressions and arrive at accurate solutions. In this case, we started with an algebraic expression, substituted given values, simplified the exponential term, and then performed multiplication and division from left to right to reach our final answer.

Key Takeaways

• The order of operations (PEMDAS) is essential for accurate calculations.
• Substituting values correctly is crucial for solving algebraic expressions.
• Exponents must be evaluated before multiplication, division, addition, and subtraction.
• Multiplication and division are performed from left to right.

In conclusion, mastering the evaluation of mathematical expressions is a fundamental skill that builds a strong foundation for further studies in mathematics and related fields. This example provides a clear illustration of the process and emphasizes the importance of accuracy and attention to detail.