What Is The Simplified Product Of The Following Expression? Assume $x \geq 0$.$\left(\sqrt{10 X^4}-x \sqrt{5 X^2}\right)\left(2 \sqrt{15 X^4}+\sqrt{3 X^3}\right$\]A. $10 X^4 \sqrt{6} + X^3 \sqrt{30 X} - 10 X^4 \sqrt{3} + X^2

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**What is the Simplified Product of the Given Expression?**

Understanding the Problem

The given expression involves the product of two terms, each containing square roots and powers of x. To simplify this expression, we need to apply the rules of algebra and manipulate the terms to obtain a simpler form.

Step 1: Multiply the Two Terms

To simplify the given expression, we start by multiplying the two terms:

(10x4−x5x2)(215x4+3x3)\left(\sqrt{10 x^4}-x \sqrt{5 x^2}\right)\left(2 \sqrt{15 x^4}+\sqrt{3 x^3}\right)

Using the distributive property, we can multiply each term in the first expression by each term in the second expression:

210x415x4+210x43x3−2x5x215x4−2x5x23x32 \sqrt{10 x^4} \sqrt{15 x^4} + 2 \sqrt{10 x^4} \sqrt{3 x^3} - 2 x \sqrt{5 x^2} \sqrt{15 x^4} - 2 x \sqrt{5 x^2} \sqrt{3 x^3}

Step 2: Simplify the Terms

Now, we can simplify each term by combining like terms and applying the rules of algebra:

210x415x4=2150x8=225x86=10x462 \sqrt{10 x^4} \sqrt{15 x^4} = 2 \sqrt{150 x^8} = 2 \sqrt{25 x^8} \sqrt{6} = 10 x^4 \sqrt{6}

210x43x3=230x72 \sqrt{10 x^4} \sqrt{3 x^3} = 2 \sqrt{30 x^7}

−2x5x215x4=−2x75x6=−2x25x63=−10x33x3-2 x \sqrt{5 x^2} \sqrt{15 x^4} = -2 x \sqrt{75 x^6} = -2 x \sqrt{25 x^6} \sqrt{3} = -10 x^3 \sqrt{3 x^3}

−2x5x23x3=−2x15x5-2 x \sqrt{5 x^2} \sqrt{3 x^3} = -2 x \sqrt{15 x^5}

Step 3: Combine Like Terms

Now, we can combine like terms to simplify the expression further:

10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}

Simplified Expression

The simplified expression is:

10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}

Q&A

Q: What is the simplified product of the given expression?

A: The simplified product of the given expression is 10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}.

Q: How do I simplify the expression?

A: To simplify the expression, you need to multiply the two terms using the distributive property, and then combine like terms.

Q: What are the rules of algebra that I need to apply?

A: You need to apply the rules of algebra, such as combining like terms and simplifying square roots.

Q: What is the final answer?

A: The final answer is 10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}.

Conclusion

In this article, we have simplified the given expression by multiplying the two terms and combining like terms. We have also provided a step-by-step guide on how to simplify the expression and answered some common questions related to the problem.

Frequently Asked Questions

  • Q: What is the simplified product of the given expression? A: The simplified product of the given expression is 10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}.
  • Q: How do I simplify the expression? A: To simplify the expression, you need to multiply the two terms using the distributive property, and then combine like terms.
  • Q: What are the rules of algebra that I need to apply? A: You need to apply the rules of algebra, such as combining like terms and simplifying square roots.
  • Q: What is the final answer? A: The final answer is 10x46+230x7−10x33x3−2x15x510 x^4 \sqrt{6} + 2 \sqrt{30 x^7} - 10 x^3 \sqrt{3 x^3} - 2 x \sqrt{15 x^5}.

Related Topics

  • Simplifying Expressions
  • Algebra Rules
  • Square Roots
  • Distributive Property

References

  • Algebra Textbook
  • Mathematics Online Resources
  • Simplifying Expressions Guide