Multiplying -8x²y⁶ And 20xy A Step-by-Step Guide With Verification

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In the realm of algebra, the multiplication of algebraic expressions is a fundamental operation. This article delves into the process of multiplying two specific algebraic expressions, -8x²y⁶ and 20xy, providing a step-by-step explanation and verification of the result. We will explore the underlying principles, demonstrate the multiplication process, and offer a clear verification method to ensure accuracy. This comprehensive guide aims to equip readers with a solid understanding of this essential algebraic skill.

Understanding the Basics of Algebraic Expression Multiplication

Before diving into the specifics of multiplying -8x²y⁶ and 20xy, let's establish a firm grasp of the fundamental principles involved in algebraic expression multiplication. At its core, this process relies on the distributive property and the rules of exponents. The distributive property, a cornerstone of algebra, states that multiplying a sum by a number is the same as multiplying each addend individually by the number and then adding the products. In simpler terms, a(b + c) = ab + ac. This property extends to expressions with multiple terms and is crucial for expanding expressions during multiplication.

Equally important are the rules of exponents. When multiplying terms with the same base, we add their exponents. This rule stems from the definition of exponents as repeated multiplication. For instance, x² * x = x^(2+1) = x³. Similarly, y⁶ * y = y^(6+1) = y⁷. Understanding these exponent rules is paramount for accurately multiplying algebraic expressions.

In essence, multiplying algebraic expressions involves combining the coefficients (the numerical parts) and applying the exponent rules to the variables. We multiply the coefficients together and then multiply the variable parts, adding the exponents of like variables. This process transforms the product into a simplified algebraic expression. Let’s say we have two algebraic terms: ax^m * by^n, where a and b are coefficients, x and y are variables, and m and n are exponents. Multiplying these terms involves first multiplying the coefficients (a * b) and then multiplying the variable parts (x^m * y^n). If x and y are the same variable, we add their exponents. This foundational knowledge paves the way for confidently tackling the multiplication of -8x²y⁶ and 20xy.

Step-by-Step Multiplication of -8x²y⁶ and 20xy

Now, let's embark on the practical process of multiplying the algebraic expressions -8x²y⁶ and 20xy. We'll break down the process into manageable steps, ensuring a clear and understandable approach. Following these steps will not only lead to the correct answer but also solidify your understanding of the underlying principles.

Step 1: Identify the Coefficients and Variables

The first step is to identify the coefficients and variables in each expression. In the expression -8x²y⁶, the coefficient is -8, the variable x has an exponent of 2 (x²), and the variable y has an exponent of 6 (y⁶). Similarly, in the expression 20xy, the coefficient is 20, the variable x has an exponent of 1 (x¹ or simply x), and the variable y has an exponent of 1 (y¹ or simply y). This identification is crucial for the subsequent multiplication steps.

Step 2: Multiply the Coefficients

The next step is to multiply the coefficients together. In this case, we multiply -8 and 20: -8 * 20 = -160. This result becomes the coefficient of the resulting expression. It's important to pay attention to the signs of the coefficients; a negative times a positive results in a negative.

Step 3: Multiply the Variables

Now, we turn our attention to the variables. We multiply the variable parts of the expressions, applying the rules of exponents. When multiplying variables with the same base, we add their exponents. Let's multiply the x terms: x² * x = x^(2+1) = x³. Next, we multiply the y terms: y⁶ * y = y^(6+1) = y⁷. This step combines the variable parts into their simplified forms.

Step 4: Combine the Results

Finally, we combine the results from Steps 2 and 3 to form the final product. We combine the coefficient (-160) with the simplified variable parts (x³ and y⁷). This yields the final result: -160x³y⁷. This expression represents the product of the two original algebraic expressions.

By following these steps meticulously, you can confidently multiply algebraic expressions. Remember, accuracy and attention to detail are key in algebra. The systematic approach outlined here will serve as a valuable tool in your algebraic endeavors.

Verification of the Result: Ensuring Accuracy

In mathematics, verification is as crucial as the calculation itself. Verifying the result of multiplying -8x²y⁶ and 20xy ensures the accuracy of our work and provides confidence in the solution. There are several methods for verification, but one straightforward approach involves substituting numerical values for the variables and comparing the results of the original expressions and the product.

Step 1: Choose Numerical Values for Variables

The first step is to choose simple numerical values for the variables x and y. Simpler values, such as 1 or 2, make calculations easier. Let's choose x = 1 and y = 1 for this verification. While any numbers can theoretically be used, using 0 or 1 can sometimes mask errors due to their special properties in multiplication and exponentiation. Therefore, slightly larger numbers are often preferable for a more robust verification.

Step 2: Substitute Values into Original Expressions

Next, we substitute these values into the original expressions, -8x²y⁶ and 20xy:

  • -8x²y⁶ = -8(1)²(1)⁶ = -8 * 1 * 1 = -8
  • 20xy = 20(1)(1) = 20 * 1 * 1 = 20

Step 3: Multiply the Results of the Original Expressions

Now, we multiply the results obtained in Step 2: -8 * 20 = -160. This value represents the expected result if our multiplication is correct.

Step 4: Substitute Values into the Product

We then substitute the same values (x = 1 and y = 1) into the product we calculated, -160x³y⁷:

  • -160x³y⁷ = -160(1)³(1)⁷ = -160 * 1 * 1 = -160

Step 5: Compare the Results

Finally, we compare the result obtained in Step 3 (-160) with the result obtained in Step 4 (-160). Since the two results are the same, this verification strongly suggests that our multiplication is correct. If the results were different, it would indicate an error in our multiplication process, prompting a re-evaluation of the steps.

This verification method provides a practical way to check the accuracy of algebraic multiplications. By substituting numerical values, we can bridge the abstract world of variables and exponents with concrete numerical calculations, making it easier to identify potential errors.

Common Mistakes to Avoid When Multiplying Algebraic Expressions

Multiplying algebraic expressions, while fundamentally straightforward, can be prone to errors if certain common pitfalls are not avoided. Recognizing these common mistakes is crucial for ensuring accuracy and developing a strong foundation in algebra. Let's explore some of the key errors to watch out for:

1. Incorrectly Applying the Distributive Property:

The distributive property is a cornerstone of algebraic multiplication, and misapplication is a frequent source of errors. Remember, the distributive property dictates that each term inside the parentheses must be multiplied by the term outside. For instance, when multiplying a(b + c), it's essential to multiply 'a' by both 'b' and 'c', resulting in ab + ac. A common mistake is to multiply 'a' only by 'b' or 'c', neglecting the other term. This incomplete distribution leads to an incorrect result. To avoid this, systematically ensure that each term within parentheses is multiplied by the term outside.

2. Errors with Exponent Rules:

The rules of exponents are fundamental to multiplying algebraic expressions, and mistakes in applying them can lead to significant errors. The most common error is forgetting to add exponents when multiplying terms with the same base. For example, x² * x should be x^(2+1) = x³, not x². Another error is incorrectly applying the power of a power rule, where (xm)n = x^(m*n). Confusing this with the multiplication rule can lead to incorrect simplification. To mitigate these errors, thoroughly review and understand the exponent rules, and practice applying them in various scenarios. A careful, step-by-step approach can help prevent these mistakes.

3. Sign Errors:

Sign errors are a pervasive issue in algebra, and multiplying algebraic expressions is no exception. The rules of sign multiplication (positive times positive equals positive, negative times negative equals positive, and positive times negative equals negative) must be meticulously applied. A common mistake is overlooking the negative sign when multiplying negative coefficients. For example, -2x * 3y should be -6xy, not 6xy. To minimize sign errors, pay close attention to the signs of each term and coefficient, and consistently apply the sign multiplication rules. Using parentheses can also help to visually separate terms and reduce the likelihood of sign errors.

4. Combining Unlike Terms Incorrectly:

In algebra, terms can only be combined if they are