Which Inequality Represents All Values Of { X $}$ For Which The Product Below Is Defined?${\sqrt{5x} \cdot \sqrt{x+4}}$A. { X \ \textgreater \ 0$}$B. { X \geq 0$}$C. { X \leq -4$} D . \[ D. \[ D . \[ X \geq

Understanding the Conditions for the Product of Square Roots to be Defined

When dealing with the product of square roots, it's essential to consider the conditions under which the expression is defined. In this case, we have the product of two square roots: $\sqrt{5x} \cdot \sqrt{x+4}$. To determine the values of $x$ for which this product is defined, we need to examine the properties of square roots and the conditions that must be met for the expression to be valid.

The Nature of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, $\sqrt{a}$ represents a value that, when squared, equals $a$. For example, $\sqrt{16}$ is equal to $4$, because $4^2 = 16$. However, not all numbers have real square roots. For instance, the square root of a negative number is not a real number.

The Conditions for the Product of Square Roots to be Defined

For the product of two square roots to be defined, each individual square root must be defined. This means that the radicand (the expression inside the square root) must be non-negative. In the case of $\sqrt{5x}$, the radicand is $5x$, which must be non-negative. Similarly, for $\sqrt{x+4}$, the radicand is $x+4$, which must also be non-negative.

Analyzing the Options

Now that we understand the conditions for the product of square roots to be defined, let's analyze the options provided:

A. $x > 0$

This option suggests that $x$ must be greater than $0$ for the product to be defined. However, this is not sufficient, as we also need to consider the condition for $\sqrt{x+4}$ to be defined.

B. $x \geq 0$

This option suggests that $x$ must be greater than or equal to $0$ for the product to be defined. However, this is still not sufficient, as we need to consider the condition for $\sqrt{x+4}$ to be defined.

C. $x \leq -4$

This option suggests that $x$ must be less than or equal to $-4$ for the product to be defined. However, this is not correct, as the radicand $x+4$ would be negative, making the square root undefined.

D. $x \geq -4$

This option suggests that $x$ must be greater than or equal to $-4$ for the product to be defined. This is the correct option, as both $5x$ and $x+4$ are non-negative when $x \geq -4$.

Conclusion

In conclusion, the correct option is D. $x \geq -4$. This is because both $5x$ and $x+4$ must be non-negative for the product of square roots to be defined. When $x \geq -4$, both conditions are met, and the product is defined.

Additional Considerations

It's worth noting that the product of square roots can be simplified using the properties of exponents. Specifically, the product of two square roots can be rewritten as a single square root of the product of the two radicands. In this case, we have:

$\sqrt{5x} \cdot \sqrt{x+4} = \sqrt{5x(x+4)}$

This simplified expression can be used to further analyze the conditions for the product to be defined.

Final Thoughts

In conclusion, the product of square roots is defined when both individual square roots are defined. In this case, we need to consider the conditions for both $5x$ and $x+4$ to be non-negative. The correct option is D. $x \geq -4$, as this ensures that both conditions are met, and the product is defined.

References

• [1] "Square Roots" by Math Open Reference. Retrieved from https://www.mathopenref.com/sqrt.html
• [2] "Properties of Square Roots" by Purplemath. Retrieved from https://www.purplemath.com/modules/radicals.htm

Table of Contents

• Understanding the Conditions for the Product of Square Roots to be Defined
• The Nature of Square Roots
• The Conditions for the Product of Square Roots to be Defined
• Analyzing the Options
• Conclusion
• Additional Considerations
• Final Thoughts
• References
• Table of Contents
  Frequently Asked Questions (FAQs) about the Product of Square Roots

In this article, we'll address some of the most common questions and concerns related to the product of square roots. Whether you're a student, a teacher, or simply someone who wants to understand the basics of square roots, this FAQ section is designed to provide you with the answers you need.

Q: What is the product of square roots?

A: The product of square roots is the result of multiplying two or more square roots together. For example, $\sqrt{5x} \cdot \sqrt{x+4}$ is the product of two square roots.

Q: What are the conditions for the product of square roots to be defined?

A: For the product of square roots to be defined, each individual square root must be defined. This means that the radicand (the expression inside the square root) must be non-negative.

Q: How do I simplify the product of square roots?

A: To simplify the product of square roots, you can use the properties of exponents. Specifically, the product of two square roots can be rewritten as a single square root of the product of the two radicands. For example, $\sqrt{5x} \cdot \sqrt{x+4} = \sqrt{5x(x+4)}$.

Q: What is the difference between a square root and a product of square roots?

A: A square root is a single value that, when multiplied by itself, gives the original number. A product of square roots, on the other hand, is the result of multiplying two or more square roots together.

Q: Can I use the product of square roots to solve equations?

A: Yes, the product of square roots can be used to solve equations. For example, if you have an equation like $\sqrt{5x} \cdot \sqrt{x+4} = 10$, you can use the product of square roots to simplify the equation and solve for $x$.

Q: What are some common mistakes to avoid when working with the product of square roots?

A: Some common mistakes to avoid when working with the product of square roots include:

• Not checking if the radicand is non-negative before taking the square root
• Not simplifying the product of square roots using the properties of exponents
• Not considering the conditions for the product of square roots to be defined

Q: How do I determine if the product of square roots is defined?

A: To determine if the product of square roots is defined, you need to check if each individual square root is defined. This means that the radicand (the expression inside the square root) must be non-negative.

Q: Can I use the product of square roots to solve inequalities?

A: Yes, the product of square roots can be used to solve inequalities. For example, if you have an inequality like $\sqrt{5x} \cdot \sqrt{x+4} > 10$, you can use the product of square roots to simplify the inequality and solve for $x$.

Q: What are some real-world applications of the product of square roots?

A: The product of square roots has many real-world applications, including:

• Calculating the area of a square or rectangle
• Finding the length of a diagonal in a square or rectangle
• Solving equations and inequalities involving square roots
• Working with quadratic equations and functions

Conclusion

In conclusion, the product of square roots is a fundamental concept in mathematics that has many real-world applications. By understanding the conditions for the product of square roots to be defined, simplifying the product of square roots using the properties of exponents, and avoiding common mistakes, you can use the product of square roots to solve equations and inequalities, and to calculate the area of a square or rectangle.

Additional Resources

• [1] "Square Roots" by Math Open Reference. Retrieved from https://www.mathopenref.com/sqrt.html
• [2] "Properties of Square Roots" by Purplemath. Retrieved from https://www.purplemath.com/modules/radicals.htm
• [3] "Solving Equations and Inequalities Involving Square Roots" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-alg-ineq/x2f-alg-ineq-sqrt

Table of Contents

• Frequently Asked Questions (FAQs) about the Product of Square Roots
• Q: What is the product of square roots?
• Q: What are the conditions for the product of square roots to be defined?
• Q: How do I simplify the product of square roots?
• Q: What is the difference between a square root and a product of square roots?
• Q: Can I use the product of square roots to solve equations?
• Q: What are some common mistakes to avoid when working with the product of square roots?
• Q: How do I determine if the product of square roots is defined?
• Q: Can I use the product of square roots to solve inequalities?
• Q: What are some real-world applications of the product of square roots?
• Conclusion
• Additional Resources
• Table of Contents