Understanding Negative Radicands

Oct 13, 2025 by magicalceylontours.com 44 views

Introduction

When dealing with expressions that involve square roots of negative numbers, we often encounter the imaginary unit $$i$$. In this article, we will explore how to simplify expressions with negative radicands in terms of $$i$$. We will use the given expression $$x = \frac{5+\sqrt{-49}}{6}$$ as an example to demonstrate the steps involved in simplifying such expressions.

Understanding Negative Radicands

A negative radicand is a number that is under a square root sign and is negative. In the given expression, $$\sqrt{-49}$$ is a negative radicand. To simplify this expression, we need to understand that $$\sqrt{-49}$$ can be rewritten as $$\sqrt{49} \cdot \sqrt{-1}$$.

Rewriting Negative Radicands

Using the property of square roots that $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$, we can rewrite $$\sqrt{-49}$$ as $$\sqrt{49} \cdot \sqrt{-1}$$.

\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1}

Since $$\sqrt{49} = 7$$, we can further simplify the expression as follows:

\sqrt{-49} = 7 \cdot \sqrt{-1}

Introducing the Imaginary Unit $$i$$

The imaginary unit $$i$$ is defined as the square root of $$-1$$. Therefore, we can rewrite $$\sqrt{-1}$$ as $$i$$.

\sqrt{-1} = i

Substituting this into the previous expression, we get:

\sqrt{-49} = 7 \cdot i

Simplifying the Original Expression

Now that we have simplified the negative radicand, we can substitute this expression back into the original expression.

x = \frac{5+\sqrt{-49}}{6}

x = \frac{5+7i}{6}

Final Simplification

To simplify the expression further, we can separate the real and imaginary parts.

x = \frac{5}{6} + \frac{7i}{6}

Conclusion

In this article, we have demonstrated how to simplify expressions with negative radicands in terms of $$i$$. We started with the given expression $$x = \frac{5+\sqrt{-49}}{6}$$ and simplified the negative radicand using the property of square roots. We then introduced the imaginary unit $$i$$ and substituted it into the expression. Finally, we simplified the expression further by separating the real and imaginary parts.

Answer

The correct answer is:

C. $$\frac{5}{6} + \frac{7i}{6}$$

Discussion

This problem requires a good understanding of the properties of square roots and the imaginary unit $$i$$. It also requires the ability to simplify complex expressions and separate the real and imaginary parts. If you have any questions or need further clarification, please feel free to ask.

Additional Resources

If you want to learn more about simplifying expressions with negative radicands in terms of $$i$$, here are some additional resources:

Khan Academy: Simplifying expressions with square roots
Mathway: Simplifying expressions with imaginary numbers
Wolfram Alpha: Simplifying expressions with negative radicands

Practice Problems

Try these practice problems to test your understanding of simplifying expressions with negative radicands in terms of $$i$$:

Simplify the expression $$x = \frac{3+\sqrt{-36}}{4}$$
Simplify the expression $$x = \frac{2+\sqrt{-25}}{3}$$
Simplify the expression $$x = \frac{4+\sqrt{-64}}{5}$$

Answer Key

$\frac{3}{4} + \frac{6i}{4}$
$\frac{2}{3} + \frac{5i}{3}$
$\frac{4}{5} + \frac{8i}{5}$$<br/>$

Introduction

In our previous article, we explored how to simplify expressions with negative radicands in terms of $$i$$. We used the given expression $$x = \frac{5+\sqrt{-49}}{6}$$ as an example to demonstrate the steps involved in simplifying such expressions. In this article, we will answer some frequently asked questions about simplifying expressions with negative radicands in terms of $$i$$.

Q&A

Q: What is a negative radicand?

A: A negative radicand is a number that is under a square root sign and is negative.

Q: How do I simplify a negative radicand?

A: To simplify a negative radicand, you can rewrite it as the product of the square root of the absolute value of the number and the imaginary unit $$i$$. For example, $$\sqrt{-49}$$ can be rewritten as $$\sqrt{49} \cdot \sqrt{-1}$$.

Q: What is the imaginary unit $$i$$?

A: The imaginary unit $$i$$ is defined as the square root of $$-1$$. It is often represented by the letter $$i$$.

Q: How do I simplify an expression with a negative radicand in terms of $$i$$?

A: To simplify an expression with a negative radicand in terms of $$i$$, you can follow these steps:

Rewrite the negative radicand as the product of the square root of the absolute value of the number and the imaginary unit $$i$$.
Substitute the rewritten negative radicand back into the original expression.
Simplify the expression further by separating the real and imaginary parts.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without using the imaginary unit $$i$$. An imaginary number is a number that can be expressed using the imaginary unit $$i$$.

Q: Can I simplify an expression with a negative radicand in terms of $$i$$ if it has multiple negative radicands?

A: Yes, you can simplify an expression with multiple negative radicands in terms of $$i$$ by following the same steps as before. You can rewrite each negative radicand as the product of the square root of the absolute value of the number and the imaginary unit $$i$$, and then substitute the rewritten negative radicands back into the original expression.

Q: Are there any special cases that I should be aware of when simplifying expressions with negative radicands in terms of $$i$$?

A: Yes, there are a few special cases that you should be aware of when simplifying expressions with negative radicands in terms of $$i$$. For example, if the negative radicand is a perfect square, you can simplify it further by taking the square root of the perfect square.