Multiplying And Verifying Algebraic Expressions -8x²y⁶ And 20xy

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In the realm of algebra, manipulating expressions is a fundamental skill. This article delves into the process of multiplying algebraic expressions, specifically focusing on the product of -8x²y⁶ and 20xy. We will meticulously break down the steps involved in this multiplication, ensuring a clear understanding of the underlying principles. Furthermore, we will verify our result by substituting specific values for the variables, providing a practical application of the concepts discussed. This exploration will enhance your ability to confidently tackle similar algebraic challenges.

Multiplying Algebraic Expressions

When multiplying algebraic expressions, the core principle is to combine like terms and apply the rules of exponents. In this case, we aim to find the product of -8x²y⁶ and 20xy. To achieve this, we will systematically multiply the coefficients and variables, carefully handling the exponents.

The process of multiplying these expressions involves several key steps:

  1. Multiply the coefficients: Begin by multiplying the numerical coefficients of the terms. In this instance, we multiply -8 and 20, resulting in -160.
  2. Multiply the x terms: Next, we multiply the x terms. We have x² in the first term and x in the second term. Recall that when multiplying variables with exponents, we add the exponents. Therefore, x² * x = x^(2+1) = x³.
  3. Multiply the y terms: Similarly, we multiply the y terms. We have y⁶ in the first term and y in the second term. Again, we add the exponents: y⁶ * y = y^(6+1) = y⁷.
  4. Combine the results: Finally, we combine the results from the previous steps to obtain the final product. This involves multiplying the coefficient, the x term, and the y term together. Thus, the product is -160x³y⁷.

Therefore, the product of -8x²y⁶ and 20xy is -160x³y⁷. This resulting expression represents the combined value of the two original expressions when multiplied together. Understanding this process is crucial for simplifying and manipulating algebraic equations.

Detailed Step-by-Step Multiplication

To further clarify the multiplication process, let's break down each step with meticulous detail:

  1. Identifying the components: We start with two expressions: -8x²y⁶ and 20xy. Each expression consists of a coefficient (the numerical part) and variables with exponents.
  2. Coefficient Multiplication: The coefficients are -8 and 20. Multiplying them gives us: -8 * 20 = -160. This is the numerical part of our final product.
  3. Multiplying the 'x' terms: We have x² and x. Remembering the rule of exponents (xᵃ * xᵇ = xᵃ⁺ᵇ), we add the exponents: x² * x = x^(2+1) = x³. The exponent of 'x' in the final product is 3.
  4. Multiplying the 'y' terms: Similarly, we have y⁶ and y. Applying the same rule of exponents: y⁶ * y = y^(6+1) = y⁷. The exponent of 'y' in the final product is 7.
  5. Combining all parts: Now, we combine the coefficient and the variable terms we calculated: -160 * x³ * y⁷ = -160x³y⁷. This gives us the final product of the multiplication.

Significance of Exponent Rules

The exponent rules are fundamental to algebraic manipulations. When multiplying terms with the same base (like 'x' or 'y'), we add their exponents. This rule stems from the basic definition of exponents as repeated multiplication. For example, x² means x * x, and x³ means x * x * x. So, x² * x³ is equivalent to (x * x) * (x * x * x), which simplifies to x⁵ (x multiplied by itself five times).

Understanding and applying these rules correctly is crucial for accurate algebraic calculations. Errors in exponent manipulation can lead to significant discrepancies in the final result. Therefore, careful attention to detail and a solid grasp of exponent rules are essential for success in algebra.

Verification of the Product

To ensure the accuracy of our result, we will verify the product by substituting the given values x = 2.5 and y = 1 into both the original expressions and the calculated product. If the values are equal, it confirms the correctness of our multiplication.

The process of verification involves the following steps:

  1. Substitute values into original expressions: First, we substitute x = 2.5 and y = 1 into the original expressions, -8x²y⁶ and 20xy, and calculate their individual values.
  2. Multiply the results: Next, we multiply the calculated values of the original expressions to obtain the product of the original expressions.
  3. Substitute values into the product: Then, we substitute x = 2.5 and y = 1 into the calculated product, -160x³y⁷, and determine its value.
  4. Compare the results: Finally, we compare the product obtained from the original expressions with the value obtained from the calculated product. If the two values are equal, it verifies the correctness of our multiplication.

Substituting Values into Original Expressions

Let's begin by substituting the given values into the original expressions:

  1. Expression 1: -8x²y⁶ Substitute x = 2.5 and y = 1: -8 * (2.5)² * (1)⁶ = -8 * 6.25 * 1 = -50 So, the value of the first expression is -50.
  2. Expression 2: 20xy Substitute x = 2.5 and y = 1: 20 * 2.5 * 1 = 50 Thus, the value of the second expression is 50.

Now that we have the values of the original expressions, we can multiply them together.

Multiplying the Results of Original Expressions

We multiply the values we obtained in the previous step:

-50 * 50 = -2500

This is the product of the original expressions when x = 2.5 and y = 1. Now, let's substitute the same values into our calculated product and see if we get the same result.

Substituting Values into the Calculated Product

Our calculated product is -160x³y⁷. Substituting x = 2.5 and y = 1 into this expression gives:

-160 * (2.5)³ * (1)⁷ = -160 * 15.625 * 1 = -2500

As we can see, the result is -2500.

Comparing the Results for Verification

We obtained the following results:

  • Product of original expressions: -2500
  • Value of calculated product: -2500

Since both values are equal, this verifies that our calculated product -160x³y⁷ is correct. This verification step is crucial in ensuring the accuracy of algebraic manipulations and provides confidence in the result.

Significance of Verification in Algebra

Verification is a critical step in algebraic problem-solving. It provides a method to check the accuracy of calculations and manipulations. By substituting specific values into both the original problem and the solution, we can confirm whether the solution holds true. This process is particularly important in complex algebraic problems where errors can easily occur.

Furthermore, verification enhances understanding. When a solution is verified, it reinforces the concepts and steps involved in the problem-solving process. It builds confidence in one's ability to apply algebraic principles correctly. In essence, verification is not just a check; it's a learning tool that strengthens mathematical skills.

In this comprehensive guide, we have explored the multiplication of algebraic expressions, focusing on the product of -8x²y⁶ and 20xy. We systematically multiplied the coefficients and variables, applying the fundamental rules of exponents. Our meticulous step-by-step approach ensured clarity and accuracy in our calculations. The resulting product was determined to be -160x³y⁷.

To validate our result, we performed a thorough verification process. By substituting the values x = 2.5 and y = 1 into both the original expressions and the calculated product, we confirmed that our multiplication was indeed correct. This verification step underscored the importance of ensuring accuracy in algebraic manipulations.

This exploration not only provided a solution to the specific problem but also reinforced key algebraic concepts. Understanding how to multiply expressions and verify results is crucial for success in algebra and related fields. With this knowledge, you are better equipped to tackle more complex algebraic challenges with confidence and precision.