What Is The Following Product? Assume $y \geq 0$.$3 \sqrt{10}\left(y^2 \sqrt{4}+\sqrt{8 Y}\right$\]A. $6 Y^2 \sqrt{10}+12 \sqrt{5 Y}$ B. $-6 \sqrt{10}+12 \sqrt{5 Y}$ C. $6 Y^2 \sqrt{10}+4 \sqrt{5 Y}$ D.

A Mathematical Expression Simplification

In this article, we will delve into the world of mathematical expressions and simplify a given product. The expression we will be working with is $3 \sqrt{10}\left(y^2 \sqrt{4}+\sqrt{8 y}\right)$. Our goal is to simplify this expression and identify the correct answer among the given options.

Understanding the Expression

The given expression is a product of two terms: $3 \sqrt{10}$ and $\left(y^2 \sqrt{4}+\sqrt{8 y}\right)$. To simplify this expression, we need to apply the rules of algebra and simplify each term separately.

Simplifying the First Term

The first term is $3 \sqrt{10}$. This term is already simplified, as it is a product of a constant and a square root.

Simplifying the Second Term

The second term is $\left(y^2 \sqrt{4}+\sqrt{8 y}\right)$. To simplify this term, we need to apply the rules of algebra. We can start by simplifying the square roots.

• $\sqrt{4}$ can be simplified as $2$, since $2^2 = 4$.
• $\sqrt{8 y}$ can be simplified as $\sqrt{4 \cdot 2 y}$, which is equal to $2 \sqrt{2 y}$.

Now, we can substitute these simplified expressions back into the second term:

$\left(y^2 \sqrt{4}+\sqrt{8 y}\right) = \left(y^2 \cdot 2 + 2 \sqrt{2 y}\right) = 2 y^2 + 2 \sqrt{2 y}$

Simplifying the Product

Now that we have simplified the second term, we can multiply it by the first term:

$3 \sqrt{10}\left(y^2 \sqrt{4}+\sqrt{8 y}\right) = 3 \sqrt{10}\left(2 y^2 + 2 \sqrt{2 y}\right)$

To simplify this expression further, we can distribute the $3 \sqrt{10}$ to each term inside the parentheses:

$3 \sqrt{10}\left(2 y^2 + 2 \sqrt{2 y}\right) = 6 y^2 \sqrt{10} + 6 \sqrt{10} \sqrt{2 y}$

Now, we can simplify the last term by combining the square roots:

$6 \sqrt{10} \sqrt{2 y} = 6 \sqrt{10 \cdot 2 y} = 6 \sqrt{20 y}$

Since $\sqrt{20} = \sqrt{4 \cdot 5} = 2 \sqrt{5}$, we can simplify the last term further:

$6 \sqrt{20 y} = 6 \cdot 2 \sqrt{5} \sqrt{y} = 12 \sqrt{5 y}$

Final Simplified Expression

Now that we have simplified the last term, we can write the final simplified expression:

$6 y^2 \sqrt{10} + 6 \sqrt{10} \sqrt{2 y} = 6 y^2 \sqrt{10} + 12 \sqrt{5 y}$

Conclusion

In this article, we simplified the given mathematical expression $3 \sqrt{10}\left(y^2 \sqrt{4}+\sqrt{8 y}\right)$ and identified the correct answer among the given options. The final simplified expression is $6 y^2 \sqrt{10} + 12 \sqrt{5 y}$.

Answer

The correct answer is A. $6 y^2 \sqrt{10} + 12 \sqrt{5 y}$

Q&A: Understanding the Simplified Expression

In this article, we will address some frequently asked questions about the simplified mathematical expression $3 \sqrt{10}\left(y^2 \sqrt{4}+\sqrt{8 y}\right)$.

Q: What is the simplified expression?

A: The simplified expression is $6 y^2 \sqrt{10} + 12 \sqrt{5 y}$.

Q: How did you simplify the expression?

A: We simplified the expression by applying the rules of algebra, including distributing the $3 \sqrt{10}$ to each term inside the parentheses and combining the square roots.

Q: What is the significance of the square roots in the expression?

A: The square roots in the expression represent the square root of the product of the terms inside the parentheses. In this case, the square roots are $\sqrt{10}$, $\sqrt{4}$, and $\sqrt{8 y}$.

Q: How do you simplify the square roots?

A: To simplify the square roots, we can use the properties of square roots, such as $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ and $\sqrt{a^2} = a$.

Q: What is the final answer?

A: The final answer is $6 y^2 \sqrt{10} + 12 \sqrt{5 y}$.

Q: How do you know which answer is correct?

A: We can verify the answer by plugging in values for $y$ and checking if the expression simplifies to the correct answer.

Q: Can you provide more examples of simplifying mathematical expressions?

A: Yes, we can provide more examples of simplifying mathematical expressions. Please let us know if you have any specific questions or topics you would like to explore.

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

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

• Not distributing the terms correctly
• Not combining the square roots correctly
• Not using the properties of square roots correctly

Q: How can I practice simplifying mathematical expressions?

A: You can practice simplifying mathematical expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and checking your work with a calculator or online tool.

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

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

• Calculating the area and perimeter of shapes
• Determining the volume of objects
• Solving equations and inequalities
• Modeling real-world phenomena

Q: Can you provide more information about the properties of square roots?

A: Yes, we can provide more information about the properties of square roots. Please let us know if you have any specific questions or topics you would like to explore.

Q: How do you know when to use the properties of square roots?

A: You can use the properties of square roots when you have a product of square roots or when you have a square root of a product. In this case, we used the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ to simplify the expression.

Q: Can you provide more examples of using the properties of square roots?

A: Yes, we can provide more examples of using the properties of square roots. Please let us know if you have any specific questions or topics you would like to explore.

Q: What are some common mistakes to avoid when using the properties of square roots?

A: Some common mistakes to avoid when using the properties of square roots include:

• Not using the correct property
• Not distributing the terms correctly
• Not combining the square roots correctly

Q: How can I practice using the properties of square roots?

A: You can practice using the properties of square roots by working through examples and exercises in your textbook or online resources. You can also try using the properties of square roots on your own and checking your work with a calculator or online tool.

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

A: Using the properties of square roots has many real-world applications, including:

• Calculating the area and perimeter of shapes
• Determining the volume of objects
• Solving equations and inequalities
• Modeling real-world phenomena

Q: Can you provide more information about the importance of simplifying mathematical expressions?

A: Yes, we can provide more information about the importance of simplifying mathematical expressions. Simplifying mathematical expressions is an essential skill in mathematics and has many real-world applications. It can help you to:

• Solve equations and inequalities
• Calculate the area and perimeter of shapes
• Determine the volume of objects
• Model real-world phenomena

Q: How can I improve my skills in simplifying mathematical expressions?

A: You can improve your skills in simplifying mathematical expressions by:

• Practicing regularly
• Working through examples and exercises in your textbook or online resources
• Using online tools and calculators to check your work
• Seeking help from a teacher or tutor if you need it

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

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

• Not distributing the terms correctly
• Not combining the square roots correctly
• Not using the properties of square roots correctly

Q: How can I stay motivated to learn and practice simplifying mathematical expressions?

A: You can stay motivated to learn and practice simplifying mathematical expressions by:

• Setting goals for yourself
• Finding real-world applications of simplifying mathematical expressions
• Working with a study group or partner
• Rewarding yourself for your progress and achievements