Simplify $6y^4(-7xy)$: A Quick Guide

When you first look at an algebraic expression like $6y^4(-7xy)$, it might seem a bit daunting, especially if you're not a math whiz. But don't worry! Simplifying expressions is a fundamental skill in algebra, and with a little practice, you'll be tackling them like a pro. Our goal today is to fully simplify the expression $6y^4(-7xy)$. This means we want to combine all the like terms and perform all possible multiplications to arrive at the most concise form of the expression. Think of it like tidying up a messy room; we want to put everything in its place so it looks neat and organized. In mathematics, this tidiness translates to a simpler, more manageable form that's easier to work with in subsequent calculations or problem-solving steps. We'll break down the process step-by-step, explaining the rules of exponents and multiplication as we go. By the end of this guide, you'll have a clear understanding of how to simplify expressions of this type, and you might even find it… dare we say… enjoyable! So, let's dive in and demystify this mathematical puzzle.

Understanding the Components of the Expression

Before we start multiplying, let's take a moment to understand what we're working with in the expression $6y^4(-7xy)$. This expression is a product of two terms: $6y^4$ and $(-7xy)$. Each term itself is composed of a coefficient and variables with exponents. The first term, $6y^4$, has a coefficient of 6 and a variable $y$ raised to the power of 4. Remember, exponents tell us how many times a number or variable is multiplied by itself. So, $y^4$ means $y \times y \times y \times y$. The second term, $(-7xy)$, has a coefficient of -7. It also has two variables, $x$ and $y$. When a variable appears without an explicit exponent, it's understood to have an exponent of 1. So, in $(-7xy)$, $x$ is actually $x^1$ and $y$ is $y^1$. Our task is to multiply these two terms together. This involves multiplying their coefficients and then combining the variables using the rules of exponents. It’s crucial to pay attention to the signs, especially the negative sign in $-7xy$, as this will affect the final sign of our simplified expression. Getting a good grasp of these basic components is the first step towards fully simplifying any algebraic expression, setting a solid foundation for the multiplication process that follows.

The Rules of Multiplication and Exponents

To fully simplify the expression $6y^4(-7xy)$, we need to apply two main mathematical rules: the rule for multiplying signed numbers and the rule for multiplying exponents with the same base. Let's tackle the coefficients first. We have a positive coefficient, 6, and a negative coefficient, -7. When you multiply a positive number by a negative number, the result is always negative. So, $6 \times (-7) = -42$. This will be the coefficient of our simplified term. Now, let's look at the variables. We have $y^4$ in the first term and $y^1$ in the second term. The rule for multiplying exponents with the same base states that when you multiply terms with the same base, you add their exponents. Therefore, $y^4 \times y^1 = y^{4+1} = y^5$. The variable $x$ only appears in the second term as $x^1$. Since there's no $x$ in the first term to multiply it with, it simply remains as $x^1$ (or just $x$) in our final expression. By correctly applying these rules – multiplying the coefficients and adding the exponents for like bases – we can systematically simplify the expression. It's like following a recipe; each step is important for the final outcome. Understanding and applying these rules correctly is key to achieving the fully simplified form we are aiming for.

Step-by-Step Simplification Process

Let's walk through the process of simplifying $6y^4(-7xy)$ step-by-step to ensure we get the fully simplified result. First, identify the coefficients and variables in each part of the expression. We have $6$ and $y^4$ in the first part, and $-7$, $x$, and $y$ in the second part. The next step is to multiply the coefficients. As we discussed, $6 \times (-7) = -42$. This gives us the numerical part of our simplified expression. Now, let's handle the variables. We have $y^4$ and $y$. Remember that $y$ is the same as $y^1$. To multiply powers of the same base, we add the exponents. So, $y^4 \times y^1 = y^{4+1} = y^5$. The variable $x$ is only present in the second term, so it remains as $x$. It's important to arrange the variables alphabetically in the final expression, so $x$ will come before $y$. Putting it all together, we combine the coefficient and the variables: $-42xy^5$. This is the fully simplified form of the original expression. Each step, from multiplying coefficients to adding exponents, is crucial. If you make a mistake in any of these steps, like forgetting the negative sign or misapplying the exponent rule, your final answer will be incorrect. This methodical approach helps ensure accuracy and leads to the most concise and correct representation of the expression.

Final Result and Verification

After carefully following the steps, we have arrived at the fully simplified expression: $-42xy^5$. To verify this, let's quickly review our work. We multiplied the coefficients $6$ and $-7$, which correctly yielded $-42$. We then combined the $y$ terms by adding their exponents: $y^4 \times y^1 = y^{4+1} = y^5$. The $x$ term, appearing only once, remained $x$. Arranging alphabetically, we get $-42xy^5$. This process adheres to the fundamental rules of algebra for multiplication and exponents. If we were to pick specific values for $x$ and $y$, we could substitute them into both the original expression and the simplified one to see if they yield the same result. For example, let $x=2$ and $y=3$.

Original expression: $6y^4(-7xy) = 6(3^4)(-7(2)(3)) = 6(81)(-42) = 486(-42) = -20412$.

Simplified expression: $-42xy^5 = -42(2)(3^5) = -42(2)(243) = -84(243) = -20412$.

Since both expressions evaluate to the same number, $-20412$, our simplification is correct. This verification method, while not always practical for complex expressions, confirms that our fully simplified answer, $-42xy^5$, is accurate and represents the original expression correctly.

Conclusion: Mastering Algebraic Simplification

Simplifying algebraic expressions like $6y^4(-7xy)$ is a core skill that unlocks further mathematical understanding and problem-solving capabilities. We’ve seen how to fully simplify this expression by systematically applying the rules of multiplication and exponents. Remember, the key steps involved multiplying the coefficients and adding the exponents of like variables. By breaking down the expression into its components and applying these rules methodically, you can confidently simplify even more complex algebraic challenges. Practice is indeed the key; the more you work through different examples, the more intuitive these processes will become. Don't shy away from reviewing the fundamental rules of exponents (like $a^m \times a^n = a^{m+n}$) and the rules for multiplying signed numbers. Mastering these foundational concepts will serve you well throughout your mathematical journey.

For further exploration and practice on algebraic simplification, you can visit reliable resources such as Khan Academy. They offer comprehensive explanations and exercises that can help reinforce your understanding of these important mathematical concepts.