Evaluating The Expression √4^2 + (3^2)/5 + 44 - (3 * 6^2) - 9 A Step-by-Step Guide

In mathematics, evaluating complex expressions requires a systematic approach, carefully following the order of operations and simplifying each step to arrive at the correct solution. In this comprehensive guide, we will walk through the process of evaluating the expression ${ \sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9} }$ step by step, providing clear explanations and insights to enhance your understanding. This article aims to help students, educators, and anyone interested in math to understand how to evaluate a complex mathematical expression. Let's dive in!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin, it’s essential to understand the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed:

1. P/B - Parentheses/Brackets: Evaluate expressions inside parentheses or brackets first.
2. E/O - Exponents/Orders: Calculate exponents and roots.
3. MD - Multiplication and Division: Perform multiplication and division from left to right.
4. AS - Addition and Subtraction: Perform addition and subtraction from left to right.

Following this order ensures that we evaluate expressions consistently and accurately. Ignoring this order can lead to incorrect results, emphasizing the importance of adhering to these rules in mathematical calculations.

Step 1: Simplify the Terms Inside the Square Root

The given expression is:

${ \sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9} }$

First, we focus on simplifying the terms inside the square root. According to PEMDAS/BODMAS, we start with exponents. The expression includes exponents such as 4^2 and 6^2, which need to be addressed first. Additionally, there is a fraction 3^2/5 and a multiplication term (3 \cdot 6^2), all within the square root that require simplification. Breaking down these elements will make the overall simplification process more manageable.

Calculating Exponents

We begin by calculating the exponents: 4^2 and 6^2. Understanding exponents is crucial as they represent the base number multiplied by itself the number of times indicated by the exponent. Accurately computing these exponential terms is fundamental to correctly evaluating the entire expression.

• 4^2 means 4 multiplied by itself: 4 \cdot 4 = 16
• 6^2 means 6 multiplied by itself: 6 \cdot 6 = 36

Substituting the Exponents

Now, substitute these values back into the expression:

${ \sqrt{16 + \frac{3^2}{5} + 44 - (3 \cdot 36) - 9} }$

Next, we need to deal with the term 3^2 in the fraction. Similar to our previous calculations, 3^2 represents 3 multiplied by itself.

• 3^2 means 3 multiplied by itself: 3 \cdot 3 = 9

Substitute this back into the expression:

${ \sqrt{16 + \frac{9}{5} + 44 - (3 \cdot 36) - 9} }$

Step 2: Perform Multiplication and Division

With the exponents handled, we move on to multiplication and division, working from left to right. The expression now includes a division (${\frac{9}{5}}$) and a multiplication (3 \cdot 36). These operations need to be performed before addition and subtraction to adhere to the order of operations.

Division: ${ \frac{9}{5} }$

First, let’s evaluate the division:

${ \frac{9}{5} = 1.8 }$

Multiplication: 3 \cdot 36

Next, perform the multiplication:

${ 3 \cdot 36 = 108 }$

Substituting the Results

Substitute these results back into the expression:

${ \sqrt{16 + 1.8 + 44 - 108 - 9} }$

Step 3: Perform Addition and Subtraction

Now, we perform addition and subtraction from left to right. This step involves combining the remaining numbers inside the square root. We’ll add and subtract the numbers in the order they appear to ensure accuracy.

Adding and Subtracting

First, add 16 and 1.8:

${ 16 + 1.8 = 17.8 }$

Next, add 17.8 and 44:

${ 17.8 + 44 = 61.8 }$

Now, subtract 108 from 61.8:

${ 61.8 - 108 = -46.2 }$

Finally, subtract 9 from -46.2:

${ -46.2 - 9 = -55.2 }$

Simplified Expression Inside the Square Root

So, the expression inside the square root simplifies to:

${ \sqrt{-55.2} }$

Step 4: Evaluate the Square Root

We have now simplified the expression inside the square root to -55.2. The final step is to evaluate the square root of this number. Understanding the nature of square roots, particularly with negative numbers, is crucial here.

Square Root of a Negative Number

The square root of a negative number is not a real number. It is an imaginary number. This is because no real number, when multiplied by itself, can result in a negative number. Therefore, we need to express the result in terms of imaginary units.

Expressing in Terms of Imaginary Units

The imaginary unit is denoted by i, where ${ i = \sqrt{-1} }$. We can rewrite the square root of -55.2 as follows:

${ \sqrt{-55.2} = \sqrt{55.2 \cdot -1} = \sqrt{55.2} \cdot \sqrt{-1} = \sqrt{55.2}i }$

Approximating the Square Root

Now, we need to approximate the square root of 55.2. The square root of 55.2 lies between the square root of 49 (which is 7) and the square root of 64 (which is 8). We can use a calculator to find a more precise value.

${ \sqrt{55.2} \approx 7.43 }$

Final Result

Therefore, the final result is:

${ \sqrt{-55.2} \approx 7.43i }$

Conclusion

In summary, evaluating the expression

${ \sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9} }$

Involves several steps that include simplifying exponents, performing multiplication and division, addition and subtraction, and finally, evaluating the square root. By following the order of operations (PEMDAS/BODMAS), we systematically reduced the expression to obtain the result ${ 7.43i }$. This process illustrates the importance of methodical calculation and understanding the properties of mathematical operations, including the handling of imaginary numbers.

This step-by-step approach not only provides the solution but also reinforces the principles of mathematical evaluation. Whether you're a student learning algebra or a math enthusiast, mastering these techniques is crucial for tackling more complex problems. Remember to always adhere to the order of operations and carefully simplify each step to ensure accuracy.

Additional Resources for Practice

To further enhance your skills in evaluating mathematical expressions, consider exploring the following resources:

• Online Math Platforms: Websites like Khan Academy, Coursera, and Udemy offer courses and practice problems on algebra and mathematical expressions.
• Textbooks and Workbooks: Many textbooks and workbooks provide detailed explanations and exercises on order of operations and algebraic simplification.
• Math Tutoring: If you need personalized help, consider working with a math tutor who can guide you through the process and provide targeted feedback.
• Practice Problems: Work through a variety of practice problems to reinforce your understanding and build confidence in your skills. Start with simpler expressions and gradually move to more complex ones.
• Math Forums and Communities: Participate in online math forums and communities where you can ask questions, discuss problems, and learn from others.

By engaging with these resources and consistently practicing, you can strengthen your mathematical skills and become proficient in evaluating complex expressions. Remember, math is a skill that improves with practice, so don’t be discouraged by challenges. Keep exploring, keep learning, and enjoy the process of mathematical discovery.