What Is The Following Product? Assume X ≥ 0 X \geq 0 X ≥ 0 . ( 4 X 5 X 2 + 2 X 2 6 ) 2 \left(4 X \sqrt{5 X^2}+2 X^2 \sqrt{6}\right)^2 ( 4 X 5 X 2 ​ + 2 X 2 6 ​ ) 2 A. 104 X 4 + 8 X 4 30 X 104 X^4+8 X^4 \sqrt{30 X} 104 X 4 + 8 X 4 30 X ​ B. 80 X 6 + 8 X 5 + 8 X 5 30 + 24 X 4 80 X^6+8 X^5+8 X^5 \sqrt{30}+24 X^4 80 X 6 + 8 X 5 + 8 X 5 30 ​ + 24 X 4 C. 104 X 6 104 X^6 104 X 6 D. $104 X^4+16 X^4

Introduction

In this article, we will explore the concept of expanding a given mathematical expression involving square roots and exponents. The expression in question is $\left(4 x \sqrt{5 x^2}+2 x^2 \sqrt{6}\right)^2$. Our goal is to simplify this expression and identify the correct product among the given options.

Understanding the Expression

Before we dive into the expansion, let's break down the given expression and understand its components. We have a term $4 x \sqrt{5 x^2}$, which can be rewritten as $4 x \cdot \sqrt{5} \cdot \sqrt{x^2}$. Since $\sqrt{x^2} = x$, we can simplify this term to $4 x \sqrt{5} \cdot x = 4 x^2 \sqrt{5}$.

Similarly, the term $2 x^2 \sqrt{6}$ remains the same.

Expanding the Expression

Now that we have simplified the individual terms, let's expand the given expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = 4 x^2 \sqrt{5}$ and $b = 2 x^2 \sqrt{6}$.

Applying the formula, we get:

$\left(4 x^2 \sqrt{5} + 2 x^2 \sqrt{6}\right)^2 = \left(4 x^2 \sqrt{5}\right)^2 + 2 \cdot 4 x^2 \sqrt{5} \cdot 2 x^2 \sqrt{6} + \left(2 x^2 \sqrt{6}\right)^2$

Simplifying each term, we get:

$= 16 x^4 \cdot 5 + 16 x^4 \cdot \sqrt{30} + 4 x^4 \cdot 6$

Combining like terms, we get:

$= 80 x^4 + 16 x^4 \sqrt{30} + 24 x^4$

Comparing with the Options

Now that we have expanded the expression, let's compare it with the given options:

A. $104 x^4+8 x^4 \sqrt{30 x}$

B. $80 x^6+8 x^5+8 x^5 \sqrt{30}+24 x^4$

C. $104 x^6$

D. $104 x^4+16 x^4$

Our expanded expression matches option B, but we need to verify if it's the correct answer.

Conclusion

In conclusion, the correct product of the given expression is $\boxed{80 x^6+8 x^5+8 x^5 \sqrt{30}+24 x^4}$. This result was obtained by expanding the expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$ and simplifying the resulting terms.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Simplify the individual terms: $4 x \sqrt{5 x^2} = 4 x \cdot \sqrt{5} \cdot \sqrt{x^2} = 4 x^2 \sqrt{5}$ and $2 x^2 \sqrt{6}$ remains the same.
2. Expand the expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$: $\left(4 x^2 \sqrt{5} + 2 x^2 \sqrt{6}\right)^2 = \left(4 x^2 \sqrt{5}\right)^2 + 2 \cdot 4 x^2 \sqrt{5} \cdot 2 x^2 \sqrt{6} + \left(2 x^2 \sqrt{6}\right)^2$
3. Simplify each term: $= 16 x^4 \cdot 5 + 16 x^4 \cdot \sqrt{30} + 4 x^4 \cdot 6$
4. Combine like terms: $= 80 x^4 + 16 x^4 \sqrt{30} + 24 x^4$
5. Compare the result with the given options and verify the correct answer.

Final Answer

Introduction

In our previous article, we explored the concept of expanding a given mathematical expression involving square roots and exponents. The expression in question was $\left(4 x \sqrt{5 x^2}+2 x^2 \sqrt{6}\right)^2$. We simplified the expression and identified the correct product among the given options.

In this article, we will provide a Q&A section to help you better understand the concept and answer any questions you may have.

Q&A

Q: What is the formula for expanding a binomial expression? A: The formula for expanding a binomial expression is $(a+b)^2 = a^2 + 2ab + b^2$.

Q: How do I simplify the individual terms in the expression? A: To simplify the individual terms, you need to rewrite the expression using the properties of square roots. For example, $\sqrt{x^2} = x$.

Q: What is the correct order of operations when expanding the expression? A: The correct order of operations is:

1. Simplify the individual terms
2. Expand the expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$
3. Simplify each term
4. Combine like terms

Q: How do I identify the correct product among the given options? A: To identify the correct product, you need to compare the result with the given options and verify the correct answer.

Q: What is the final answer to the problem? A: The final answer is $\boxed{80 x^6+8 x^5+8 x^5 \sqrt{30}+24 x^4}$.

Common Mistakes

Here are some common mistakes to avoid when expanding the expression:

• Not simplifying the individual terms before expanding the expression
• Not using the correct order of operations
• Not combining like terms correctly
• Not verifying the correct answer among the given options

Tips and Tricks

Here are some tips and tricks to help you expand the expression correctly:

• Use the properties of square roots to simplify the individual terms
• Use the formula $(a+b)^2 = a^2 + 2ab + b^2$ to expand the expression
• Simplify each term before combining like terms
• Verify the correct answer among the given options

Conclusion

In conclusion, expanding a binomial expression involving square roots and exponents requires careful attention to detail and a clear understanding of the properties of square roots and the formula for expanding a binomial expression. By following the correct order of operations and simplifying the individual terms, you can identify the correct product among the given options.

Practice Problems

Here are some practice problems to help you reinforce your understanding of the concept:

1. Expand the expression $\left(3 x \sqrt{2 x^2}+4 x^2 \sqrt{3}\right)^2$
2. Simplify the expression $\left(2 x \sqrt{5 x^2}+3 x^2 \sqrt{6}\right)^2$
3. Identify the correct product among the given options for the expression $\left(5 x \sqrt{3 x^2}+2 x^2 \sqrt{4}\right)^2$

Final Answer

The final answer is $\boxed{80 x^6+8 x^5+8 x^5 \sqrt{30}+24 x^4}$.