Simplifying $\frac{\sqrt{-63}}{\sqrt{7}}$: A Step-by-Step Guide

Introduction

In mathematics, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying the expression ${\frac{\sqrt{-63}}{\sqrt{7}}}$ . This particular expression involves complex numbers due to the presence of a negative value under the square root. To effectively simplify this, we need to understand how to handle imaginary units and radicals. This comprehensive guide will walk you through each step, ensuring clarity and a solid understanding of the underlying principles. Mastering this type of simplification not only enhances your mathematical proficiency but also builds a strong foundation for more advanced topics in algebra and complex analysis. So, let’s embark on this mathematical journey and break down the process of simplifying ${\frac{\sqrt{-63}}{\sqrt{7}}}$ in a clear and concise manner. We’ll cover everything from identifying imaginary numbers to rationalizing denominators, making sure you’re equipped to tackle similar problems with confidence.

Understanding Imaginary Numbers

Before we dive into the simplification process, it's crucial to understand imaginary numbers. Imaginary numbers arise when we take the square root of a negative number. The imaginary unit, denoted as i, is defined as ${ i = \sqrt{-1} }$. This concept is pivotal in dealing with expressions like ${\sqrt{-63}}$. To properly grasp this, remember that the square root of a negative number cannot be a real number, hence the introduction of i. When you encounter a negative number under a square root, the first step is to express it in terms of i. For instance, ${\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a}i}$ where a is a positive real number. This understanding forms the bedrock of simplifying expressions involving complex numbers. We need to recognize that i is not just a symbol but a number that follows specific rules, such as ${ i^2 = -1 }$. Keeping this in mind will prevent common mistakes and ensure accurate simplification. Moreover, familiarity with imaginary numbers is essential for various branches of mathematics and physics, making it a valuable tool in your problem-solving arsenal. As we move forward, you'll see how this concept is applied practically in simplifying our target expression.

Expressing ${\sqrt{-63}}$ in terms of i

To express ${\sqrt{-63}}$ in terms of i, we first recognize that -63 can be written as ${63 \times -1}$. Therefore, ${\sqrt{-63}}$ can be rewritten as ${\sqrt{63 \times -1}}$. Using the property of square roots that ${\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}}$, we can separate this into ${\sqrt{63} \cdot \sqrt{-1}}$. Since ${\sqrt{-1}}$ is defined as i, we have ${\sqrt{63} \cdot i}$. Now, we need to simplify ${\sqrt{63}}$. We look for perfect square factors of 63. The factors of 63 are 1, 3, 7, 9, 21, and 63. We see that 9 is a perfect square and 63 can be written as ${9 \times 7}$. Thus, ${\sqrt{63}}$ can be further simplified to ${\sqrt{9 \times 7}}$, which equals ${\sqrt{9} \cdot \sqrt{7}}$. Since ${\sqrt{9} = 3}$, we have ${3\sqrt{7}}$. Putting it all together, ${\sqrt{-63}}$ becomes ${3\sqrt{7}i}$. This step is crucial because it transforms the expression into a form where we can easily perform further simplifications. Remember, breaking down the square root into its simplest terms allows for a clearer and more manageable calculation.

Simplifying the Expression

Now that we have expressed ${\sqrt{-63}}$ in terms of i, we can proceed with simplifying the entire expression, which is ${\frac{\sqrt{-63}}{\sqrt{7}}}$. We've determined that ${\sqrt{-63} = 3\sqrt{7}i}$, so we can substitute this into the original expression, giving us ${\frac{3\sqrt{7}i}{\sqrt{7}}}$. The next step involves canceling out common factors. In this case, we have ${\sqrt{7}}$ in both the numerator and the denominator. By canceling ${\sqrt{7}}$, we simplify the expression to ${3i}$. This simplification demonstrates a fundamental principle in mathematics: identifying and canceling common factors to arrive at a simpler form. The ability to spot these factors quickly can significantly streamline the problem-solving process. After this cancellation, the expression is reduced to its simplest form, which is ${3i}$. This result highlights the elegance and efficiency of mathematical simplification, where complex-looking expressions can often be reduced to surprisingly simple forms. By following these steps, you can confidently tackle similar problems and gain a deeper appreciation for the structure and beauty of mathematical expressions.

Substituting and Canceling Common Factors

Substituting ${\sqrt{-63}}$ with ${3\sqrt{7}i}$ in the original expression ${\frac{\sqrt{-63}}{\sqrt{7}}}$ gives us ${\frac{3\sqrt{7}i}{\sqrt{7}}}$. The critical next step is to identify common factors in the numerator and the denominator. Here, we can clearly see that ${\sqrt{7}}$ appears in both. When we have a common factor, we can cancel it out to simplify the expression. Think of it as dividing both the numerator and the denominator by the same value, which doesn't change the overall value of the expression. So, by canceling ${\sqrt{7}}$ from both the numerator and the denominator, we are left with ${\frac{3i}{1}}$, which is simply ${3i}$. This cancellation simplifies the expression dramatically and brings us closer to the final answer. It's a fundamental technique in simplifying fractions and algebraic expressions. Recognizing and acting on these common factors is a skill that becomes second nature with practice and is essential for efficient problem-solving in mathematics.

Final Result

After performing the necessary steps, we arrive at the final simplified result: ${3i}$. This means that ${\frac{\sqrt{-63}}{\sqrt{7}}}$ simplifies to the imaginary number ${3i}$. This result underscores the importance of understanding complex numbers and how to manipulate them correctly. The process involved breaking down the square root of a negative number, expressing it in terms of i, and then simplifying the resulting expression by canceling common factors. The final answer, ${3i}$, is a pure imaginary number, meaning it has no real part. This is a common outcome when dealing with square roots of negative numbers in the numerator and real numbers in the denominator. The ability to arrive at this simplified form is a testament to the power of algebraic manipulation and a solid understanding of complex number properties. This result not only answers the specific problem but also reinforces the broader principles of mathematical simplification and the handling of imaginary numbers. It's a concise and elegant solution that highlights the beauty of mathematics in reducing complex expressions to their simplest forms.

Conclusion

In conclusion, simplifying ${\frac{\sqrt{-63}}{\sqrt{7}}}$ involves several key steps: understanding imaginary numbers, expressing ${\sqrt{-63}}$ in terms of i, substituting this expression back into the original equation, and canceling common factors. By following these steps carefully, we have shown that ${\frac{\sqrt{-63}}{\sqrt{7}} = 3i}$. This exercise not only provides a solution to a specific problem but also reinforces essential mathematical concepts and techniques. The ability to simplify complex expressions is a crucial skill in mathematics, applicable in various fields ranging from algebra to calculus. Mastering these techniques builds confidence and proficiency in mathematical problem-solving. Remember, each step is built upon fundamental principles, such as the definition of i and the properties of square roots and fractions. By understanding these principles, you can approach similar problems with ease and accuracy. Keep practicing, and you'll find that simplifying complex expressions becomes second nature. This article has provided a detailed walkthrough, and with continued effort, you'll be well-equipped to tackle mathematical challenges involving complex numbers and radicals.

For further exploration of complex numbers and their properties, consider visiting Khan Academy's Complex Numbers section.