What Is The Following Product? Assume $x \geq 0$. ( 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 + 88 X 5 30 + 24 X 4 80 X^6+8 X^5 + 88 X^5 \sqrt{30}+24 X^4 80 X 6 + 8 X 5 + 88 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. The expression 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 answer among the given options.

Understanding the Expression

The given expression involves two terms: $4 x \sqrt{5 x^2}$ and $2 x^2 \sqrt{6}$. To simplify this expression, we need to expand it using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = 4 x \sqrt{5 x^2}$ and $b = 2 x^2 \sqrt{6}$.

Expanding the Expression

To expand the expression, we need to square each term and then multiply them together. Let's start by squaring the first term:

$$\left(4 x \sqrt{5 x^2}\right)^2 = \left(4 x\right)^2 \left(\sqrt{5 x^2}\right)^2 = 16 x^2 \cdot 5 x^2 = 80 x^4$$

Next, we need to square the second term:

$$\left(2 x^2 \sqrt{6}\right)^2 = \left(2 x^2\right)^2 \left(\sqrt{6}\right)^2 = 4 x^4 \cdot 6 = 24 x^4$$

Now, we need to multiply the two squared terms together:

$$80 x^4 \cdot 24 x^4 = 1920 x^8$$

However, we are not done yet. We also need to multiply the two terms together using the formula $2ab$:

$$2 \cdot 4 x \sqrt{5 x^2} \cdot 2 x^2 \sqrt{6} = 16 x^3 \sqrt{5 x^2 \cdot 6} = 16 x^3 \sqrt{30 x^2}$$

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by combining like terms:

$$80 x^4 + 24 x^4 + 16 x^3 \sqrt{30 x^2}$$

However, we can simplify this expression further by factoring out a common term:

$$104 x^4 + 16 x^4 \sqrt{30 x}$$

Conclusion

In conclusion, the correct answer is $\boxed{104 x^4+8 x^4 \sqrt{30 x}}$. This expression is the result of expanding the given expression $\left(4 x \sqrt{5 x^2}+2 x^2 \sqrt{6}\right)^2$.

Discussion

The given expression involves a combination of square roots and exponents. To simplify this expression, we need to expand it using the formula $(a+b)^2 = a^2 + 2ab + b^2$. This requires us to square each term and then multiply them together.

Key Takeaways

• To simplify an expression involving square roots and exponents, we need to expand it using the formula $(a+b)^2 = a^2 + 2ab + b^2$.
• We need to square each term and then multiply them together.
• We can simplify the expression by combining like terms and factoring out a common term.

Final Answer

Introduction

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

Q&A Session

In this article, we will answer some frequently asked questions related to the given expression and its simplification.

Q: What is the formula for expanding an expression involving square roots?

A: The formula for expanding an expression involving square roots is $(a+b)^2 = a^2 + 2ab + b^2$.

Q: How do we square each term in the expression?

A: To square each term, we need to multiply the term by itself. For example, to square the term $4 x \sqrt{5 x^2}$, we need to multiply it by itself: $\left(4 x \sqrt{5 x^2}\right)^2 = \left(4 x\right)^2 \left(\sqrt{5 x^2}\right)^2 = 16 x^2 \cdot 5 x^2 = 80 x^4$.

Q: How do we multiply the two squared terms together?

A: To multiply the two squared terms together, we need to multiply them using the formula $2ab$. For example, to multiply the two terms $80 x^4$ and $24 x^4$, we need to multiply them using the formula $2ab$: $2 \cdot 4 x \sqrt{5 x^2} \cdot 2 x^2 \sqrt{6} = 16 x^3 \sqrt{5 x^2 \cdot 6} = 16 x^3 \sqrt{30 x^2}$.

Q: How do we simplify the expression?

A: To simplify the expression, we need to combine like terms and factor out a common term. For example, to simplify the expression $80 x^4 + 24 x^4 + 16 x^3 \sqrt{30 x^2}$, we can factor out a common term: $104 x^4 + 16 x^4 \sqrt{30 x}$.

Q: What is the final answer?

A: The final answer is $\boxed{104 x^4+8 x^4 \sqrt{30 x}}$.

Common Mistakes

When simplifying expressions involving square roots, there are several common mistakes to avoid:

• Not expanding the expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$.
• Not squaring each term correctly.
• Not multiplying the two squared terms together correctly.
• Not simplifying the expression by combining like terms and factoring out a common term.

Tips and Tricks

When simplifying expressions involving square roots, here are some tips and tricks to keep in mind:

• Make sure to expand the expression using the formula $(a+b)^2 = a^2 + 2ab + b^2$.
• Square each term correctly by multiplying it by itself.
• Multiply the two squared terms together using the formula $2ab$.
• Simplify the expression by combining like terms and factoring out a common term.

Conclusion

In conclusion, simplifying expressions involving square roots requires careful attention to detail and a thorough understanding of the formulas and techniques involved. By following the steps outlined in this article, you can simplify even the most complex expressions and arrive at the correct answer.

Final Answer

The final answer is $\boxed{104 x^4+8 x^4 \sqrt{30 x}}$.