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 +

Mar 9, 2025 by ADMIN 399 views

**What is the Following Product?**

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 answer 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 simplified as $4 x \cdot \sqrt{5} \cdot x$ . This is because the square root of $x^2$ is simply $x$ . Additionally, we have the term $2 x^2 \sqrt{6}$ .

Expanding the Expression

To expand the given expression, we will use the formula $(a + b)^2 = a^2 + 2ab + b^2$ . In this case, $a = 4 x \sqrt{5 x^2}$ and $b = 2 x^2 \sqrt{6}$ .

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

Simplifying the Expression

Now, let's simplify each term in the expanded expression.

= \left(4 x \sqrt{5} \cdot x\right)^2 + 2 \cdot 4 x \sqrt{5} \cdot x \cdot 2 x^2 \sqrt{6} + \left(2 x^2 \sqrt{6}\right)^2
= \left(4 x \sqrt{5} \cdot x\right)^2 + 2 \cdot 4 x \sqrt{5} \cdot x \cdot 2 x^2 \sqrt{6} + 4 x^4 \cdot 6
= 16 x^2 \cdot 5 x^2 + 16 x^2 \cdot 2 x^2 \cdot 2 x^2 \sqrt{6} + 24 x^4
= 80 x^4 + 64 x^6 \sqrt{6} + 24 x^4

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of $\sqrt{6}$ is $\sqrt{6}$ .

= 80 x^4 + 64 x^6 \sqrt{6} + 24 x^4
= 104 x^4 + 64 x^6 \sqrt{6}

However, we can further simplify the expression by multiplying the numerator and denominator by $\sqrt{6}$ .

= 104 x^4 + 64 x^6 \sqrt{6}
= 104 x^4 + 64 x^6 \sqrt{6} \cdot \frac{\sqrt{6}}{\sqrt{6}}
= 104 x^4 + 64 x^6 \cdot \frac{6}{\sqrt{6} \cdot \sqrt{6}}
= 104 x^4 + 64 x^6 \cdot \frac{6}{6}
= 104 x^4 + 64 x^6

Conclusion

In conclusion, the correct answer is B. $80 x^6 + 8 x^5 + 8 x^5 \sqrt{30} + 24 x^4$ . This is because the expanded expression can be simplified as $80 x^6 + 8 x^5 + 8 x^5 \sqrt{30} + 24 x^4$ .

Discussion

The given expression involves the expansion of a binomial squared, which is a common technique in algebra. The expression also involves the use of square roots and exponents, which can be simplified using various mathematical identities.

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 $\left(4 x \sqrt{5 x^2} + 2 x^2 \sqrt{6}\right)^2$ . Our goal was to simplify this expression and identify the correct answer among the given options.

Q&A Session

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

Q: How do we simplify the expression $4 x \sqrt{5 x^2}$ ? A: We can simplify the expression $4 x \sqrt{5 x^2}$ as $4 x \cdot \sqrt{5} \cdot x$ .

Q: What is the conjugate of $\sqrt{6}$ ? A: The conjugate of $\sqrt{6}$ is $\sqrt{6}$ .

Q: How do we rationalize the denominator of the expression $64 x^6 \sqrt{6}$ ? A: We can rationalize the denominator of the expression $64 x^6 \sqrt{6}$ by multiplying the numerator and denominator by $\sqrt{6}$ .

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

Common Mistakes

Not using the correct formula for expanding a binomial squared.
Not simplifying the expression correctly.
Not rationalizing the denominator correctly.

Tips and Tricks

Make sure to use the correct formula for expanding a binomial squared.
Simplify the expression step by step.
Rationalize the denominator correctly.

Conclusion

In conclusion, the Q&A session provides a summary of the key concepts and formulas used to simplify the given expression. It also highlights common mistakes and provides tips and tricks for simplifying expressions involving square roots and exponents.

Final Answer

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

Discussion

Additional Resources

For more information on expanding binomials, visit Math Open Reference.
For more information on simplifying expressions involving square roots and exponents, visit Khan Academy.

Final Thoughts

Simplifying expressions involving square roots and exponents requires a deep understanding of mathematical concepts and formulas. By following the correct steps and using the correct formulas, we can simplify even the most complex expressions.