The Product $\left(\frac{2xy}{z}\right)\left(\frac{2x^2}{3yz}\right$\] Is:A. $\frac{4x^3}{3z^2}$ B. $\frac{4x^3}{3z}$ C. $\frac{4x^2}{2z}$ D. $\frac{4x^2}{3z^2}$

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Understanding the Problem

The given problem involves the multiplication of two algebraic expressions. We are required to simplify the product and identify the correct answer from the given options.

The Algebraic Expressions

The two algebraic expressions are:

(2xyz)\left(\frac{2xy}{z}\right) and (2x23yz)\left(\frac{2x^2}{3yz}\right)

Multiplying the Expressions

To find the product of these two expressions, we need to multiply the numerators and denominators separately.

(2xyz)(2x23yz)=(2xy)(2x2)(z)(3yz)\left(\frac{2xy}{z}\right)\left(\frac{2x^2}{3yz}\right) = \frac{(2xy)(2x^2)}{(z)(3yz)}

Simplifying the Product

Now, let's simplify the product by combining like terms in the numerator and canceling out common factors in the denominator.

(2xy)(2x2)(z)(3yz)=4x3y3z2y\frac{(2xy)(2x^2)}{(z)(3yz)} = \frac{4x^3y}{3z^2y}

Canceling Out Common Factors

We can see that the variable yy appears in both the numerator and denominator. We can cancel out these common factors to simplify the expression further.

4x3y3z2y=4x33z2\frac{4x^3y}{3z^2y} = \frac{4x^3}{3z^2}

Conclusion

Therefore, the product of the two algebraic expressions is 4x33z2\frac{4x^3}{3z^2}.

Answer

The correct answer is:

A. 4x33z2\frac{4x^3}{3z^2}

Final Thoughts

In this problem, we learned how to multiply two algebraic expressions and simplify the product by combining like terms and canceling out common factors. This is an important skill in algebra and is used extensively in various mathematical applications.

Key Takeaways

  • To multiply two algebraic expressions, multiply the numerators and denominators separately.
  • Simplify the product by combining like terms in the numerator and canceling out common factors in the denominator.
  • Cancel out common factors in the numerator and denominator to simplify the expression further.

Common Mistakes to Avoid

  • Failing to multiply the numerators and denominators separately.
  • Not simplifying the product by combining like terms and canceling out common factors.
  • Not canceling out common factors in the numerator and denominator.

Real-World Applications

The concept of multiplying algebraic expressions and simplifying the product is used extensively in various mathematical applications, such as:

  • Calculus: The product rule and quotient rule in calculus involve multiplying and dividing algebraic expressions.
  • Physics: In physics, the product of two or more variables is often used to describe physical quantities, such as force and acceleration.
  • Engineering: In engineering, the product of two or more variables is often used to describe physical quantities, such as stress and strain.

Conclusion

In conclusion, the product of the two algebraic expressions is 4x33z2\frac{4x^3}{3z^2}. This problem demonstrates the importance of multiplying and simplifying algebraic expressions in various mathematical applications. By following the steps outlined in this problem, we can simplify complex algebraic expressions and arrive at the correct solution.

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Q: What is the first step in multiplying two algebraic expressions?

A: The first step in multiplying two algebraic expressions is to multiply the numerators and denominators separately.

Q: How do I simplify the product of two algebraic expressions?

A: To simplify the product of two algebraic expressions, combine like terms in the numerator and cancel out common factors in the denominator.

Q: What is the difference between multiplying and dividing algebraic expressions?

A: Multiplying algebraic expressions involves multiplying the numerators and denominators separately, while dividing algebraic expressions involves dividing the numerator by the denominator.

Q: Can I cancel out common factors in the numerator and denominator when multiplying algebraic expressions?

A: Yes, you can cancel out common factors in the numerator and denominator when multiplying algebraic expressions. This will simplify the expression and make it easier to work with.

Q: What is the product rule in calculus?

A: The product rule in calculus is a rule for finding the derivative of a product of two functions. It involves multiplying the two functions and then taking the derivative of the result.

Q: How do I apply the product rule in calculus?

A: To apply the product rule in calculus, you need to multiply the two functions and then take the derivative of the result. This involves using the chain rule and the product rule for differentiation.

Q: What is the quotient rule in calculus?

A: The quotient rule in calculus is a rule for finding the derivative of a quotient of two functions. It involves dividing the two functions and then taking the derivative of the result.

Q: How do I apply the quotient rule in calculus?

A: To apply the quotient rule in calculus, you need to divide the two functions and then take the derivative of the result. This involves using the chain rule and the quotient rule for differentiation.

Q: Can I use the product rule and quotient rule in physics and engineering?

A: Yes, you can use the product rule and quotient rule in physics and engineering to describe physical quantities and solve problems.

Q: How do I apply the product rule and quotient rule in physics and engineering?

A: To apply the product rule and quotient rule in physics and engineering, you need to multiply or divide the relevant variables and then take the derivative of the result. This involves using the chain rule and the product rule or quotient rule for differentiation.

Q: What are some common mistakes to avoid when multiplying algebraic expressions?

A: Some common mistakes to avoid when multiplying algebraic expressions include failing to multiply the numerators and denominators separately, not simplifying the product by combining like terms and canceling out common factors, and not canceling out common factors in the numerator and denominator.

Q: How can I practice multiplying algebraic expressions?

A: You can practice multiplying algebraic expressions by working through examples and exercises in a textbook or online resource. You can also try solving problems on your own and then checking your answers with a calculator or online tool.

Q: What are some real-world applications of multiplying algebraic expressions?

A: Some real-world applications of multiplying algebraic expressions include calculus, physics, and engineering. In these fields, the product rule and quotient rule are used to describe physical quantities and solve problems.

Q: Can I use algebraic expressions in other areas of mathematics?

A: Yes, you can use algebraic expressions in other areas of mathematics, such as geometry and trigonometry. In these fields, algebraic expressions are used to describe geometric shapes and trigonometric functions.

Q: How do I apply algebraic expressions in geometry and trigonometry?

A: To apply algebraic expressions in geometry and trigonometry, you need to use algebraic expressions to describe geometric shapes and trigonometric functions. This involves using variables and algebraic operations to represent geometric and trigonometric quantities.

Q: What are some common algebraic expressions used in geometry and trigonometry?

A: Some common algebraic expressions used in geometry and trigonometry include the Pythagorean theorem, the distance formula, and the trigonometric identities.

Q: How can I practice applying algebraic expressions in geometry and trigonometry?

A: You can practice applying algebraic expressions in geometry and trigonometry by working through examples and exercises in a textbook or online resource. You can also try solving problems on your own and then checking your answers with a calculator or online tool.