Multiply And Simplify Algebraic Expression (8/5 Xy)(1/3 X^2 Y)(-30 X^2 Y)

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In mathematics, simplifying expressions is a fundamental skill. This article delves into the process of multiplying and simplifying algebraic expressions, focusing on an example involving fractions and variables. We will walk through the steps, highlighting key concepts and techniques to ensure a clear understanding.

Understanding the Basics of Algebraic Expressions

Before we tackle the main problem, let's establish a solid foundation. Algebraic expressions are combinations of variables, constants, and mathematical operations. Variables, typically represented by letters such as x and y, stand for unknown values. Constants are fixed numerical values, like 8, 5, 1, 3, and -30 in our example. The operations we'll be dealing with include multiplication, division, and exponentiation.

When multiplying algebraic expressions, we apply the commutative and associative properties of multiplication. The commutative property allows us to change the order of factors without affecting the product (e.g., a * b = b * a). The associative property allows us to regroup factors without changing the product (e.g., (a * b) * c = a * (b * c)). These properties are crucial for rearranging and simplifying expressions efficiently. Furthermore, we need to remember the rules of exponents: when multiplying terms with the same base, we add the exponents (e.g., x^m * x^n = x^(m+n)). This rule is essential for combining variable terms in our expression. Understanding these basics will make the process of multiplying and simplifying algebraic expressions much more manageable. Remember, practice is key, so working through various examples will help solidify your understanding and build your confidence in algebraic manipulations.

Problem Statement: Multiplying and Simplifying the Expression

Our main task is to multiply and simplify the expression:

(85xy)(13x2y)(−30x2y)(\frac{8}{5} x y)(\frac{1}{3} x^2 y)(-30 x^2 y)

This expression involves three terms, each containing fractions, variables (x and y), and exponents. Our goal is to combine these terms into a single, simplified expression. This requires careful application of the rules of multiplication and exponents. We'll proceed step-by-step, first by rearranging the terms, then multiplying the coefficients (the numerical parts), and finally combining the variable terms. Each step is crucial for arriving at the correct simplified form. By breaking down the problem into smaller, manageable parts, we can avoid errors and gain a clearer understanding of the process. This problem is a great example of how algebraic expressions can be manipulated and simplified, a skill that's fundamental in many areas of mathematics and its applications.

Step-by-Step Solution

1. Rearrange the Terms

Using the commutative and associative properties of multiplication, we can rearrange the terms to group the coefficients together and the variables together:

(85∗13∗−30)∗(x∗x2∗x2)∗(y∗y∗y)(\frac{8}{5} * \frac{1}{3} * -30) * (x * x^2 * x^2) * (y * y * y)

This rearrangement makes the multiplication process more organized and easier to follow. By grouping similar terms, we can focus on each part separately, reducing the chance of making mistakes. The commutative property allows us to change the order of the factors, and the associative property allows us to regroup them without changing the result. This step is a crucial setup for the subsequent multiplication steps.

2. Multiply the Coefficients

Now, let's multiply the coefficients:

85∗13∗−30=8∗1∗−305∗3=−24015=−16\frac{8}{5} * \frac{1}{3} * -30 = \frac{8 * 1 * -30}{5 * 3} = \frac{-240}{15} = -16

Here, we multiply the numerators together and the denominators together, then simplify the resulting fraction. This step involves basic arithmetic operations with fractions. It's important to pay attention to the signs (positive and negative) to ensure the correct result. In this case, the product of the coefficients is -16. This numerical part will be a key component of our final simplified expression.

3. Multiply the 'x' Terms

Next, we multiply the x terms:

x∗x2∗x2=x1+2+2=x5x * x^2 * x^2 = x^{1+2+2} = x^5

Remember, when multiplying terms with the same base, we add the exponents. Here, x is equivalent to x^1. So, we add the exponents 1, 2, and 2 to get 5. The result is x^5. This step demonstrates a fundamental rule of exponents, which is crucial for simplifying algebraic expressions involving variables.

4. Multiply the 'y' Terms

Similarly, we multiply the y terms:

y∗y∗y=y1+1+1=y3y * y * y = y^{1+1+1} = y^3

Again, we add the exponents. Each y term is equivalent to y^1. Adding the exponents 1, 1, and 1 gives us 3. The result is y^3. This step mirrors the process used for the x terms, reinforcing the rule of exponents and its application in simplifying algebraic expressions.

5. Combine the Results

Finally, we combine the results from steps 2, 3, and 4:

−16∗x5∗y3=−16x5y3-16 * x^5 * y^3 = -16x^5y^3

This step brings together the numerical coefficient and the variable terms to form the final simplified expression. We simply multiply the coefficient (-16) by the x term (x^5) and the y term (y^3). The result, -16x^5y^3, is the simplified form of the original expression. This final step demonstrates how all the individual simplifications come together to give the complete answer.

Final Simplified Expression

Therefore, the simplified expression is:

−16x5y3-16x^5y^3

This is the final answer, representing the product of the original terms in its simplest form. We have successfully multiplied and simplified the given algebraic expression by applying the commutative and associative properties of multiplication, the rules of exponents, and basic arithmetic operations. This result is a single term that encapsulates the combined effect of the original three terms.

Key Concepts and Takeaways

  • Commutative and Associative Properties: These properties allow us to rearrange and regroup terms, making multiplication easier.
  • Rules of Exponents: When multiplying terms with the same base, add the exponents (x^m * x^n = x^(m+n)).
  • Step-by-Step Approach: Breaking down the problem into smaller steps helps avoid errors and makes the process more manageable.
  • Combining Like Terms: Grouping coefficients and variables separately simplifies the multiplication process.

Understanding these concepts and practicing these techniques will greatly improve your ability to multiply and simplify algebraic expressions. The step-by-step approach ensures clarity and reduces the likelihood of mistakes, while the underlying mathematical principles provide a solid foundation for more complex algebraic manipulations. This skill is essential for success in various mathematical fields, making it a worthwhile investment of your time and effort.

Practice Problems

To further solidify your understanding, try these practice problems:

  1. (3a2b)(−2ab3)(4a3)(3a^2b)(-2ab^3)(4a^3)
  2. (12xy2)(6x3y)(−23x)(\frac{1}{2}xy^2)(6x^3y)(-\frac{2}{3}x)
  3. (−5p2q3)(2pq2)(−p3)(-5p^2q^3)(2pq^2)(-p^3)

Working through these problems will provide valuable practice in applying the concepts and techniques discussed in this article. Each problem presents a slightly different challenge, encouraging you to adapt your approach and reinforce your understanding. Remember to follow the step-by-step process outlined earlier, and don't hesitate to review the explanations and examples if needed. The key to mastering algebraic manipulation is consistent practice and a thorough understanding of the underlying principles.

By working through these examples and practice problems, you'll gain confidence in your ability to multiply and simplify algebraic expressions. This is a crucial skill in algebra and other areas of mathematics, so consistent practice is key.