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

Evaluating mathematical expressions can sometimes feel like navigating a complex maze, especially when square roots, fractions, exponents, and multiple operations are involved. This article provides a step-by-step guide to simplify and solve the expression ${\sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9}}$. We will break down each component, ensuring a clear understanding of the process. Whether you are a student looking to improve your math skills or someone who enjoys mathematical challenges, this guide will help you tackle similar problems with confidence. Grasping the order of operations and understanding how to simplify expressions is crucial in mathematics, paving the way for more advanced concepts. By the end of this article, you will not only be able to solve this particular expression but also gain a broader understanding of how to approach and simplify complex mathematical problems. The journey through this problem will enhance your problem-solving skills and deepen your appreciation for the elegance of mathematical operations. Let’s embark on this mathematical adventure together and unravel the solution! This intricate expression requires careful attention to detail and a thorough understanding of mathematical principles. Each step, from addressing exponents and fractions to performing arithmetic operations, is crucial for arriving at the correct answer. The process involves applying the order of operations (PEMDAS/BODMAS), which dictates the sequence in which calculations must be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Neglecting this order can lead to significant errors, highlighting the importance of methodical execution. Furthermore, familiarity with how to handle fractions and square roots is essential, as these components add layers of complexity to the expression. The ability to simplify fractions and accurately calculate square roots is fundamental in mathematics. Ultimately, this guide aims not only to provide the solution but also to foster a deeper understanding of the underlying mathematical concepts. By dissecting the problem and explaining each step, we empower readers to approach similar challenges with greater confidence and skill. This holistic approach ensures that the knowledge gained extends beyond the specific problem at hand, building a solid foundation for future mathematical endeavors.

Step-by-Step Solution

To effectively evaluate the expression ${\sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9}}$, we need to follow the order of operations meticulously. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures that we perform calculations in the correct sequence. Our initial focus will be on simplifying the terms inside the square root. This involves addressing exponents first, then performing multiplication, followed by addition and subtraction. By breaking down the problem into smaller, manageable steps, we can avoid errors and maintain clarity throughout the process. Each step is crucial, and a clear understanding of the underlying principles is essential for success. The initial evaluation of exponents will transform the expression into a more workable form, setting the stage for subsequent calculations. Similarly, handling the multiplication term accurately is vital, as it directly affects the final outcome. Once these operations are completed, we can move on to the addition and subtraction, further simplifying the expression. By adhering to the order of operations, we maintain the integrity of the mathematical process and ensure that our final answer is accurate. This meticulous approach not only helps in solving this particular expression but also reinforces fundamental mathematical skills applicable to a wide range of problems. Let's begin by addressing the exponents and then systematically work through the remaining operations, keeping a clear and organized approach throughout. This methodical execution is key to successfully navigating the complexities of the expression and arriving at the correct solution. The principles we apply here are universal in mathematics, underscoring the importance of mastering these fundamental concepts.

1. Evaluate Exponents

The first step in simplifying the expression is to evaluate the exponents. We have two exponential terms: ${4^2}$ and ${6^2}$. Let's calculate these:

• ${4^2 = 4 \cdot 4 = 16}$
• ${6^2 = 6 \cdot 6 = 36}$

Now, let's also address the ${3^2}$ in the fraction:

• ${3^2 = 3 \cdot 3 = 9}$

Substituting these values back into the original expression, we get:

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

Evaluating exponents is a fundamental step in simplifying mathematical expressions. It transforms the terms into manageable numbers, paving the way for subsequent operations. The accurate computation of these exponents is crucial for the overall correctness of the solution. In this case, squaring the numbers simplifies the expression significantly, making it easier to proceed with multiplication, addition, and subtraction. The ability to correctly handle exponents is a cornerstone of mathematical proficiency. Understanding that ${a^2}$ means ${a \cdot a}$ is essential for grasping more complex mathematical concepts. This step demonstrates the importance of following the order of operations, where exponents are addressed before other arithmetic operations. By tackling exponents first, we reduce the complexity of the expression and set the stage for a more streamlined simplification process. The accuracy achieved in this step directly impacts the accuracy of the final result, highlighting the need for precision and attention to detail. This foundational understanding of exponents and their role in mathematical expressions is invaluable for any aspiring mathematician or student. The simplification process initiated here underscores the power of breaking down complex problems into smaller, more manageable parts, a strategy that is applicable across various mathematical disciplines. This step is not just about calculating squares; it’s about instilling a systematic approach to problem-solving.

2. Perform Multiplication

Next, we need to address the multiplication within the parentheses: ${3 \cdot 36}$.

• ${3 \cdot 36 = 108}$

Substituting this result back into the expression, we have:

${\sqrt{16 + \frac{9}{5} + 44 - 108 - 9}}$

Performing multiplication accurately is a critical step in simplifying mathematical expressions. In this instance, multiplying 3 by 36 gives us 108, which is a significant value that impacts the overall result. The correct execution of this step ensures that we maintain the integrity of the expression and move closer to the final solution. Multiplication is a fundamental arithmetic operation, and its accurate application is essential for mathematical proficiency. This step highlights the importance of paying attention to the details of the problem and following the order of operations diligently. By addressing multiplication before addition and subtraction, we adhere to established mathematical conventions and avoid potential errors. The result of this multiplication directly influences the subsequent addition and subtraction operations, underscoring its significance. Understanding how to perform multiplication efficiently and accurately is a valuable skill in mathematics. This step not only simplifies the expression but also demonstrates the interconnectedness of mathematical operations. The ability to correctly multiply numbers is a building block for more advanced mathematical concepts. The methodical approach employed here is crucial for navigating complex expressions and arriving at the correct answer. This emphasis on precision and accuracy is a hallmark of effective mathematical problem-solving. By completing this multiplication step, we bring the expression closer to its simplified form, preparing it for the final stages of evaluation.

3. Handle the Fraction

Now, let’s handle the fraction ${\frac{9}{5}}$. To make it easier to combine with the other terms, we can convert it to a decimal:

• ${\frac{9}{5} = 1.8}$

Substituting this back into the expression, we get:

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

Handling fractions is a crucial skill in mathematics, and in this step, we address the fraction ${\frac{9}{5}}$. Converting the fraction to a decimal makes it easier to perform addition and subtraction with the other terms in the expression. The conversion of ${\frac{9}{5}}$ to 1.8 is a straightforward process, but it underscores the importance of understanding the relationship between fractions and decimals. This step simplifies the expression and allows us to combine like terms more readily. The ability to work with fractions and decimals interchangeably is a valuable asset in mathematical problem-solving. This conversion also highlights the flexibility in mathematical notation and the importance of choosing the most convenient form for the given problem. By transforming the fraction into a decimal, we enhance the clarity of the expression and reduce the likelihood of errors in subsequent calculations. This step demonstrates the practical application of mathematical principles in simplifying complex problems. The accuracy in converting fractions to decimals is essential, as any error here will propagate through the rest of the solution. The methodical approach to handling fractions, as demonstrated in this step, is a key component of mathematical fluency. By incorporating this step into our solution, we ensure a smoother and more efficient path to the final answer. This process reinforces the understanding that fractions and decimals are different representations of the same numerical values, and the choice of representation can significantly impact the ease of calculation.

4. Perform Addition and Subtraction

Next, we perform the addition and subtraction operations inside the square root, working from left to right:

• ${16 + 1.8 = 17.8}$
• ${17.8 + 44 = 61.8}$
• ${61.8 - 108 = -46.2}$
• ${-46.2 - 9 = -55.2}$

So the expression inside the square root simplifies to -55.2:

${\sqrt{-55.2}}$

Performing addition and subtraction accurately is a fundamental step in simplifying mathematical expressions. In this step, we combine the terms inside the square root, working from left to right, to ensure we follow the correct order of operations. The sequence of operations is crucial: first, we add 16 and 1.8 to get 17.8. Then, we add 44 to 17.8, resulting in 61.8. Next, we subtract 108 from 61.8, which gives us -46.2. Finally, we subtract 9 from -46.2, resulting in -55.2. This meticulous process underscores the importance of attention to detail and adherence to the order of operations. The accuracy in each of these calculations is vital for arriving at the correct final result. Addition and subtraction are basic arithmetic operations, but their correct application is essential in complex expressions like this one. The negative sign introduced in this step highlights the importance of understanding how to work with negative numbers. By performing these operations systematically, we reduce the complexity of the expression and pave the way for the final step. This step not only simplifies the expression but also reinforces the concept that addition and subtraction are inverse operations. The methodical approach used here demonstrates the value of breaking down a complex problem into smaller, more manageable parts. By carefully executing each addition and subtraction operation, we ensure the integrity of the mathematical process and move closer to the final solution.

5. Evaluate the Square Root

Now, we need to evaluate the square root of -55.2:

${\sqrt{-55.2}}$

Since we are taking the square root of a negative number, the result is an imaginary number. We can express this as:

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

Where ${i}$ is the imaginary unit, defined as ${i = \sqrt{-1}}$.

We can approximate ${\sqrt{55.2}}$ as approximately 7.43.

Therefore, the final result is approximately:

${7.43i}$

Evaluating the square root is a crucial final step in solving our expression. Here, we encounter a square root of a negative number, which introduces the concept of imaginary numbers. The square root of -55.2 is not a real number, as there is no real number that, when multiplied by itself, yields a negative result. This leads us to the imaginary unit, denoted by ${i}$, where ${i = \sqrt{-1}}$. By expressing ${\sqrt{-55.2}}$ as ${\sqrt{55.2} \cdot \sqrt{-1}}$, we can separate the real and imaginary components. This step demonstrates the extension of the number system beyond real numbers to include complex numbers, which have both real and imaginary parts. The approximation of ${\sqrt{55.2}}$ as 7.43 provides a practical value for the magnitude of the imaginary component. This process underscores the importance of understanding the properties of square roots and the nature of negative numbers within the context of square roots. The final result, approximately 7.43${i}$, is an imaginary number, highlighting the importance of recognizing when a mathematical expression leads to non-real solutions. This step showcases the power of mathematical abstraction and the ability to work with numbers beyond the realm of everyday counting. The introduction of imaginary numbers expands the scope of mathematical problem-solving and opens the door to more advanced concepts in algebra and calculus. By correctly handling the square root of a negative number, we complete the simplification of the expression and arrive at a meaningful mathematical result.

Final Answer

The expression ${\sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9}}$ simplifies to approximately ${7.43i}$.

This final answer encapsulates the journey of simplifying a complex mathematical expression, starting from the initial application of the order of operations to the eventual emergence of an imaginary number. The meticulous step-by-step process has not only provided the solution but also highlighted the importance of precision and understanding in mathematics. By correctly evaluating exponents, handling fractions, performing multiplication, and executing addition and subtraction, we navigated the expression to its simplified form. The encounter with the square root of a negative number led us to the realm of imaginary numbers, demonstrating the richness and complexity of the number system. The final result, approximately 7.43${i}$, stands as a testament to the power of mathematical principles and the ability to solve intricate problems through systematic approaches. This comprehensive solution underscores the value of breaking down complex tasks into manageable steps and adhering to established mathematical conventions. The journey from the initial expression to the final answer reinforces the importance of each individual operation and the interconnectedness of mathematical concepts. This entire process serves as a valuable learning experience, enhancing problem-solving skills and fostering a deeper appreciation for the beauty and rigor of mathematics. The final answer is not just a number; it represents the culmination of careful calculations and a thorough understanding of mathematical principles.

Conclusion

In conclusion, evaluating the expression ${\sqrt{4^2 + \frac{3^2}{5} + 44 - (3 \cdot 6^2) - 9}}$ requires a methodical approach, adhering strictly to the order of operations. Each step, from evaluating exponents to handling fractions and performing arithmetic operations, is crucial for arriving at the correct answer. The final result, approximately ${7.43i}$, is an imaginary number, which arises from taking the square root of a negative value. This problem exemplifies the importance of understanding fundamental mathematical concepts and their application in complex scenarios. The process of simplification not only provides a solution but also enhances problem-solving skills and reinforces mathematical intuition. By breaking down the expression into smaller, manageable parts, we can tackle even the most daunting mathematical challenges. The journey through this problem highlights the interconnectedness of various mathematical operations and the significance of precision in calculations. The emergence of an imaginary number as the final result underscores the richness and complexity of the mathematical landscape. This exercise serves as a valuable reminder of the power of systematic thinking and the rewards of mastering mathematical principles. The confidence gained from successfully navigating this problem can be applied to a wide range of mathematical endeavors, fostering a deeper appreciation for the subject. Ultimately, this evaluation demonstrates that mathematics is not just about numbers and symbols; it’s about logical reasoning, problem-solving, and the pursuit of accurate and meaningful solutions.