What Is The Product? Assume $x \geq 0$.$(\sqrt{3x} + \sqrt{5})(\sqrt{15x} + 2\sqrt{30}$\]A. $3x\sqrt{5} + 3\sqrt{165x} + 10\sqrt{6}$B. $3x\sqrt{5} + 6\sqrt{10x} + 5\sqrt{3x} + 10\sqrt{6}$C. $3x\sqrt{5} +

Understanding the Problem

The given problem involves finding the product of two algebraic expressions, each containing square roots. The expressions are $(\sqrt{3x} + \sqrt{5})$ and $(\sqrt{15x} + 2\sqrt{30})$. To find the product, we need to apply the distributive property and simplify the resulting expression.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can apply this property to the given expressions by multiplying each term in the first expression by each term in the second expression.

$(\sqrt{3x} + \sqrt{5})(\sqrt{15x} + 2\sqrt{30})$

$= (\sqrt{3x})(\sqrt{15x}) + (\sqrt{3x})(2\sqrt{30}) + (\sqrt{5})(\sqrt{15x}) + (\sqrt{5})(2\sqrt{30})$

Step 2: Simplify the Expression

Now, we can simplify each term in the expression.

$= \sqrt{45x^2} + 2\sqrt{90x} + \sqrt{75x} + 2\sqrt{150}$

$= \sqrt{9 \cdot 5 \cdot x^2} + 2\sqrt{9 \cdot 10 \cdot x} + \sqrt{25 \cdot 3 \cdot x} + 2\sqrt{25 \cdot 6}$

$= 3x\sqrt{5} + 2 \cdot 3 \sqrt{10x} + 5\sqrt{3x} + 2 \cdot 5 \sqrt{6}$

$= 3x\sqrt{5} + 6\sqrt{10x} + 5\sqrt{3x} + 10\sqrt{6}$

Conclusion

The product of the two given expressions is $3x\sqrt{5} + 6\sqrt{10x} + 5\sqrt{3x} + 10\sqrt{6}$. This is the correct answer.

Comparison with Options

Let's compare our answer with the given options.

A. $3x\sqrt{5} + 3\sqrt{165x} + 10\sqrt{6}$

B. $3x\sqrt{5} + 6\sqrt{10x} + 5\sqrt{3x} + 10\sqrt{6}$

C. $3x\sqrt{5} + 10\sqrt{6}$

Our answer matches option B.

Final Answer

The final answer is $\boxed{B}$.

Discussion

Frequently Asked Questions

In the previous article, we discussed how to find the product of two algebraic expressions, each containing square roots. Here are some frequently asked questions and their answers to help you better understand the concept.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to expand expressions by multiplying each term in one expression by each term in another expression.

Q: How do I apply the distributive property to find the product of two expressions?

A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression. This will result in a new expression that contains multiple terms. You can then simplify the expression by combining like terms.

Q: What is the difference between the product of two expressions and the sum of two expressions?

A: The product of two expressions is the result of multiplying each term in one expression by each term in the other expression. The sum of two expressions is the result of adding each term in one expression to each term in the other expression.

Q: How do I simplify an expression that contains square roots?

A: To simplify an expression that contains square roots, you need to combine like terms and simplify the square roots. You can do this by multiplying the square roots together and simplifying the resulting expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{B}$, which is $3x\sqrt{5} + 6\sqrt{10x} + 5\sqrt{3x} + 10\sqrt{6}$.

Common Mistakes to Avoid

When finding the product of two algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not applying the distributive property: Make sure to multiply each term in one expression by each term in the other expression.
• Not simplifying the expression: Combine like terms and simplify the square roots to get the final answer.
• Not checking the answer: Make sure to compare your answer with the given options to ensure that you have the correct answer.

Conclusion

Finding the product of two algebraic expressions can be a challenging task, but with the right techniques and strategies, you can simplify the expression and get the final answer. Remember to apply the distributive property, simplify the expression, and check the answer to ensure that you have the correct solution.

Additional Resources

If you're struggling with finding the product of two algebraic expressions, here are some additional resources that may help:

• Online tutorials: Websites like Khan Academy and Mathway offer video tutorials and interactive exercises to help you learn how to find the product of two algebraic expressions.
• Practice problems: Practice problems are a great way to reinforce your understanding of the concept. Try solving problems on your own or using online resources to get help.
• Math textbooks: Math textbooks often have chapters and exercises dedicated to finding the product of two algebraic expressions. Consult a math textbook for more information.

Final Thoughts

Finding the product of two algebraic expressions is an important concept in mathematics. By understanding the distributive property and simplifying expressions, you can solve problems and get the final answer. Remember to avoid common mistakes and use additional resources to help you learn.