Simplifying Negative Numbers under Square Roots

Oct 13, 2025 by magicalceylontours.com 128 views

Understanding the Problem

When dealing with complex expressions, it's essential to simplify them to make them easier to work with. In this article, we'll focus on simplifying the expression $\sqrt{-100}$ and determining which of the given choices is equivalent to it.

The Expression: $\sqrt{-100}$

The expression $\sqrt{-100}$ involves the square root of a negative number. To simplify this expression, we need to understand the properties of square roots and how they interact with negative numbers.

Simplifying Negative Numbers under Square Roots

When we take the square root of a negative number, we can rewrite it as the square root of a complex number. In this case, $\sqrt{-100}$ can be rewritten as $\sqrt{(-1)(100)}$ . This can be further simplified to $\sqrt{-1}\sqrt{100}$ .

The Square Root of -1

The square root of -1 is a fundamental concept in mathematics, denoted by the imaginary unit $i$ . By definition, $i$ is the square root of -1, and it satisfies the equation $i^2 = -1$ .

Simplifying the Expression

Now that we've simplified the expression $\sqrt{-100}$ to $\sqrt{-1}\sqrt{100}$ , we can further simplify it by evaluating the square root of 100. The square root of 100 is 10, so we can rewrite the expression as $i\sqrt{100}$ .

Evaluating the Final Expression

Now that we've simplified the expression to $i\sqrt{100}$ , we can evaluate the final result. The square root of 100 is 10, so the final expression is $i\cdot 10 = 10i$ .

Comparing the Results

Now that we've simplified the expression $\sqrt{-100}$ to $10i$ , we can compare it to the given choices. The correct answer is:

E. $10i$

Conclusion

In this article, we simplified the expression $\sqrt{-100}$ by rewriting it as the square root of a complex number. We then evaluated the square root of -1 and simplified the expression to $10i$ . By comparing the results to the given choices, we determined that the correct answer is E. $10i$ .

Key Takeaways

The square root of a negative number can be rewritten as the square root of a complex number.
The square root of -1 is denoted by the imaginary unit $i$ .
Simplifying complex expressions involves understanding the properties of square roots and how they interact with negative numbers.

Additional Resources

For more information on simplifying complex expressions, check out the following resources:

Khan Academy: Simplifying Complex Expressions
Mathway: Simplifying Complex Expressions
Wolfram Alpha: Simplifying Complex Expressions

Final Thoughts

Q: What is the square root of a negative number?

A: The square root of a negative number is a complex number. It can be rewritten as the square root of a complex number, where the complex number is in the form of $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify the square root of a negative number?

A: To simplify the square root of a negative number, you can rewrite it as the square root of a complex number. This involves breaking down the negative number into its real and imaginary parts. For example, $\sqrt{-100}$ can be rewritten as $\sqrt{(-1)(100)}$ , which can be further simplified to $\sqrt{-1}\sqrt{100}$ .

Q: What is the square root of -1?

A: The square root of -1 is denoted by the imaginary unit $i$ . By definition, $i$ is the square root of -1, and it satisfies the equation $i^2 = -1$ .

Q: How do I simplify the expression $\sqrt{-100}$ ?

A: To simplify the expression $\sqrt{-100}$ , you can follow these steps:

Rewrite the expression as $\sqrt{(-1)(100)}$ .
Simplify the expression to $\sqrt{-1}\sqrt{100}$ .
Evaluate the square root of 100, which is 10.
Simplify the expression to $i\sqrt{100}$ .
Evaluate the final expression, which is $10i$ .

Q: What is the correct answer for the expression $\sqrt{-100}$ ?

A: The correct answer for the expression $\sqrt{-100}$ is $10i$ .

Q: How do I determine which choice is equivalent to the given expression?

A: To determine which choice is equivalent to the given expression, you can follow these steps:

Simplify the expression to its final form.
Compare the simplified expression to the given choices.
Select the choice that is equivalent to the simplified expression.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

Not rewriting the negative number as a complex number.
Not breaking down the negative number into its real and imaginary parts.
Not evaluating the square root of the complex number correctly.
Not simplifying the expression to its final form.

Q: What are some additional resources for learning about simplifying complex expressions?

A: Some additional resources for learning about simplifying complex expressions include:

Khan Academy: Simplifying Complex Expressions
Mathway: Simplifying Complex Expressions
Wolfram Alpha: Simplifying Complex Expressions

Q: How can I practice simplifying complex expressions?

A: You can practice simplifying complex expressions by working through examples and exercises. Some ways to practice include:

Working through practice problems in a textbook or online resource.
Creating your own practice problems and solving them.
Joining a study group or working with a tutor to practice simplifying complex expressions.

Q: What are some real-world applications of simplifying complex expressions?

A: Simplifying complex expressions has many real-world applications, including:

Calculating the roots of a polynomial equation.
Finding the maximum or minimum value of a function.
Solving systems of linear equations.
Calculating the area or perimeter of a complex shape.