Solving Complex Numbers Expressing √-16 + √49 In A + Bi Form

Navigating the realm of complex numbers can seem daunting at first, but with a clear understanding of the fundamentals, even seemingly intricate problems become manageable. In this comprehensive guide, we will dissect the complex number expression $\sqrt{-16} + \sqrt{49}$, step by step, to express it in the standard $a + bi$ form. This exploration will not only provide the solution but also enhance your grasp of complex number operations and their significance in mathematics. Let’s dive in and unravel the mystery behind this expression.

Breaking Down the Components: $\sqrt{-16}$ and $\sqrt{49}$

To effectively tackle the expression $\sqrt{-16} + \sqrt{49}$, we must first deconstruct it into its individual components. This approach allows us to address each part separately and then combine the results. Let’s begin with the first term, $\sqrt{-16}$. The key to understanding this term lies in recognizing the presence of the negative sign inside the square root. In the realm of real numbers, the square root of a negative number is undefined. This is where the concept of imaginary numbers comes into play.

Unveiling the Imaginary Unit: i

The imaginary unit, denoted by i, is defined as the square root of -1, that is, i = $\sqrt{-1}$. This seemingly simple definition opens up a whole new dimension in mathematics, allowing us to work with square roots of negative numbers. Armed with this knowledge, we can rewrite $\sqrt{-16}$ as follows:

$\sqrt{-16} = \sqrt{16 \times -1} = \sqrt{16} \times \sqrt{-1}$

Now, we know that $\sqrt{16}$ is 4, and $\sqrt{-1}$ is i. Therefore:

$\sqrt{-16} = 4i$

This transformation is a crucial step in expressing the given expression in the desired $a + bi$ form. By isolating the imaginary unit, we can treat it as a separate component and combine it appropriately with the real part of the expression.

Tackling the Real Part: $\sqrt{49}$

The second term in our expression, $\sqrt{49}$, is much more straightforward. It is simply the square root of a positive number, which we can easily calculate. The square root of 49 is 7, as 7 multiplied by itself equals 49. Therefore:

$\sqrt{49} = 7$

This real number will form the 'a' part of our complex number in the $a + bi$ form. With both components now simplified, we are ready to combine them and express the original expression in its complex form.

Combining Real and Imaginary Parts: Expressing in $a + bi$ Form

Now that we have simplified both components of the expression, $\sqrt{-16}$ and $\sqrt{49}$, we can combine them to express the original expression in the standard complex number form, $a + bi$. We found that $\sqrt{-16} = 4i$ and $\sqrt{49} = 7$. Therefore, we can rewrite the original expression as:

$\sqrt{-16} + \sqrt{49} = 4i + 7$

To express this in the standard $a + bi$ form, we simply rearrange the terms so that the real part comes first, followed by the imaginary part:

$7 + 4i$

This is the complex number in the desired form, where a is 7 and b is 4. The real part of the complex number is 7, and the imaginary part is 4i. This final step demonstrates how to combine real and imaginary numbers to form a complex number, which is a fundamental concept in complex number arithmetic.

Identifying the Correct Answer: A Comprehensive Review

Having successfully expressed $\sqrt{-16} + \sqrt{49}$ in the complex number form $a + bi$, we have arrived at the solution: $7 + 4i$. Now, let's review the given options to identify the correct answer and discuss why the other options are incorrect. This step is crucial for reinforcing our understanding and avoiding common pitfalls.

Evaluating the Options

The original question presented us with four options:

A. $7 - 4i$ B. $7 + 4i$ C. $-4 + 7i$ D. $4 + 7i$

By meticulously breaking down the problem and performing the necessary calculations, we arrived at the complex number $7 + 4i$. Comparing this result with the given options, we can clearly see that option B, $7 + 4i$, is the correct answer. The other options represent common errors that can occur when working with complex numbers, such as incorrect handling of the imaginary unit or misplacing the real and imaginary parts.

Understanding the Incorrect Options

Let's briefly discuss why the other options are incorrect:

• A. $7 - 4i$: This option is incorrect because it has a negative sign in front of the imaginary part. This would be the result if we had calculated $\sqrt{49} - \sqrt{-16}$ instead of $\sqrt{-16} + \sqrt{49}$.
• C. $-4 + 7i$: This option seems to have confused the real and imaginary parts and also incorrectly assigned a negative sign to the real part. This is a common mistake, highlighting the importance of carefully tracking the signs and components.
• D. $4 + 7i$: This option correctly identifies the magnitudes of the real and imaginary parts but incorrectly swaps their positions. It demonstrates the importance of understanding the standard $a + bi$ form, where the real part comes first.

By understanding why these options are incorrect, we can solidify our understanding of complex number operations and avoid making similar mistakes in the future.

Key Takeaways: Mastering Complex Number Operations

In this comprehensive exploration, we have successfully navigated the complex number expression $\sqrt{-16} + \sqrt{49}$ and expressed it in the standard $a + bi$ form. This journey has highlighted several key concepts and techniques essential for mastering complex number operations. Let's recap these takeaways to reinforce your understanding.

The Significance of the Imaginary Unit

The imaginary unit, i, defined as $\sqrt{-1}$, is the cornerstone of complex numbers. It allows us to work with square roots of negative numbers, expanding the realm of mathematics beyond real numbers. Remember, i squared (i^2) is equal to -1, a crucial property when simplifying complex expressions.

Breaking Down Complex Expressions

Complex expressions often appear daunting at first glance. However, by breaking them down into smaller, manageable components, we can tackle them systematically. In this case, we separated $\sqrt{-16} + \sqrt{49}$ into $\sqrt{-16}$ and $\sqrt{49}$, simplified each part individually, and then combined the results.

Expressing in $a + bi$ Form

The standard form for a complex number is $a + bi$, where a represents the real part and bi represents the imaginary part. Expressing complex numbers in this form allows for easy comparison and arithmetic operations. Ensure you correctly identify and position the real and imaginary parts when writing a complex number in this form.

Avoiding Common Pitfalls

Working with complex numbers requires careful attention to detail. Common mistakes include misinterpreting the imaginary unit, incorrectly handling signs, and swapping the real and imaginary parts. By understanding these pitfalls and practicing diligently, you can avoid these errors and confidently navigate complex number problems.

By mastering these key takeaways, you will be well-equipped to tackle a wide range of complex number problems. Remember, practice is key to solidifying your understanding and building confidence in your mathematical abilities.

Conclusion: Embracing the World of Complex Numbers

In conclusion, we have successfully solved the complex number expression $\sqrt{-16} + \sqrt{49}$ and expressed it in the standard $a + bi$ form, which is $7 + 4i$. This process has not only provided the solution but also illuminated the fundamental concepts and techniques involved in complex number operations. By understanding the imaginary unit, breaking down complex expressions, and adhering to the standard $a + bi$ form, we can confidently navigate the world of complex numbers.

Complex numbers play a vital role in various fields of mathematics, physics, and engineering. From electrical circuit analysis to quantum mechanics, their applications are vast and significant. By mastering the basics, you are opening doors to a deeper understanding of these advanced topics. So, embrace the world of complex numbers, continue practicing, and unlock the endless possibilities they offer.