Simplifying Algebraic Expressions Step By Step

by ADMIN 47 views

Hey guys! Let's dive into simplifying algebraic expressions. It might seem tricky at first, but with a clear approach, it becomes quite manageable. Today, we'll break down a specific problem step by step, so you can feel confident tackling similar questions. We're going to take the expression $-3 x y^3[4 x^2 y-7 x y^2)$ and simplify it. Let’s make algebra less of a headache and more of a breeze!

Breaking Down the Problem

Understanding the Expression

Before we start simplifying, let's take a good look at the expression: $-3 x y^3[4 x^2 y-7 x y^2)$. This expression involves a monomial (a single term) being multiplied by a binomial (an expression with two terms inside the parentheses). The monomial is $-3xy^3$, and the binomial is $(4x^2y - 7xy^2)$. Our goal here is to eliminate the parentheses by distributing the monomial across each term within the binomial. This is a classic application of the distributive property, which is a fundamental concept in algebra. Grasping this property is essential because it's used extensively in more complex algebraic manipulations. Essentially, we need to multiply $-3xy^3$ by both $4x^2y$ and $-7xy^2$.

The Distributive Property: Our Key Tool

The distributive property states that $a(b + c) = ab + ac$. In simpler terms, it means we multiply the term outside the parentheses by each term inside the parentheses. This property is the backbone of our simplification process. By applying this, we ensure that every term within the binomial is correctly accounted for when combined with the monomial. For our expression, this translates to multiplying $-3xy^3$ by $4x^2y$ and then multiplying $-3xy^3$ by $-7xy^2$. We’re not just blindly following a rule; we’re carefully expanding the expression to a form that’s easier to work with. Understanding why we use the distributive property is as important as knowing how to use it.

Identifying Like Terms

One crucial aspect of simplifying expressions is recognizing and combining like terms. Like terms are terms that have the same variables raised to the same powers. For example, $3x^2y$ and $-5x^2y$ are like terms because they both have $x^2y$. However, $3x^2y$ and $3xy^2$ are not like terms because the powers of $x$ and $y$ are different. Recognizing like terms allows us to simplify expressions by adding or subtracting their coefficients (the numbers in front of the variables). In our expression, after we distribute, we'll need to check for like terms to see if we can simplify further. If we find terms like $x3y4$ and $-5x3y4$, we can combine them into $-4x3y4$. This step is crucial for achieving the most simplified form of the expression.

Step-by-Step Simplification

Step 1: Distribute the Monomial

Okay, let's get our hands dirty and start simplifying. Remember, we're distributing $-3xy^3$ across the binomial $(4x^2y - 7xy^2)$. This means we perform two multiplications:

  1. −3xy3∗4x2y-3xy^3 * 4x^2y

  2. −3xy3∗−7xy2-3xy^3 * -7xy^2

Let’s take each multiplication separately to avoid confusion.

Step 2: Multiply the First Term

First, we multiply $-3xy^3$ by $4x^2y$. When multiplying terms with exponents, we multiply the coefficients (the numbers) and add the exponents of like variables. So, we have:

  • Coefficients: $-3 * 4 = -12$

  • x$ variables: $x^1 * x^2 = x^{1+2} = x^3

  • y$ variables: $y^3 * y^1 = y^{3+1} = y^4

Combining these, we get $-12x3y4$. Make sure you're comfortable with these exponent rules; they're essential for simplifying algebraic expressions. We’ve successfully handled the first part of the distribution. Now, onto the second part!

Step 3: Multiply the Second Term

Next, we multiply $-3xy^3$ by $-7xy^2$. Again, we'll handle the coefficients and variables separately:

  • Coefficients: $-3 * -7 = 21$

  • x$ variables: $x^1 * x^1 = x^{1+1} = x^2

  • y$ variables: $y^3 * y^2 = y^{3+2} = y^5

Putting it all together, we get $21x2y5$. Notice how a negative times a negative gives us a positive. This is a common area for mistakes, so always double-check your signs!

Step 4: Combine the Results

Now that we've multiplied $-3xy^3$ by both terms in the binomial, we combine the results. We found that:

  • −3xy3∗4x2y=−12x3y4-3xy^3 * 4x^2y = -12x^3y^4

  • −3xy3∗−7xy2=21x2y5-3xy^3 * -7xy^2 = 21x^2y^5

So, the simplified expression is $-12x3y4 + 21x2y5$. This is the result of distributing and multiplying. The next step is to see if we can simplify further by combining like terms.

Step 5: Check for Like Terms

Here's where we pause and ask: Are there any like terms in our simplified expression, $-12x3y4 + 21x2y5$? Remember, like terms have the same variables raised to the same powers. Looking closely, we see that the first term has $x3y4$, and the second term has $x2y5$. The exponents on the $x$ and $y$ are different in each term, meaning they are not like terms. Since there are no like terms to combine, our expression is already in its simplest form. This step is crucial because sometimes you might think you're done, but there might be further simplification possible.

Final Answer and Conclusion

The Simplified Expression

After distributing, multiplying, and checking for like terms, we've arrived at our final simplified expression: $-12x3y4 + 21x2y5$. This means that the original expression, $-3 x y^3(4 x^2 y-7 x y^2)$, simplifies to $-12x3y4 + 21x2y5$. This result matches option E in the given choices.

Wrapping Up

So, the correct answer is E. $-12 x^3 y^4+21 x^2 y^5$. We took a seemingly complex algebraic expression and broke it down into manageable steps. By applying the distributive property, carefully multiplying terms, and checking for like terms, we successfully simplified the expression. Remember, practice makes perfect! The more you work with these concepts, the more comfortable you’ll become. Keep at it, guys, and you’ll master these algebraic manipulations in no time!

Practice Problems

Test Your Understanding

To really nail down these skills, let's look at a couple of practice problems. These are similar to the one we just worked through, so you can apply the same strategies. Working through these will help you identify any areas where you might need a bit more practice.

  1. Simplify: $2a2b(3ab2 - 5a^2b)$
  2. Simplify: $-4pq3(2p2q - 6pq^2)$

Try solving these on your own. If you get stuck, revisit the steps we discussed earlier. Remember, the key is to distribute carefully, multiply the coefficients and variables correctly, and combine like terms if any.

Solutions and Explanations

After you’ve tried the problems, check your answers against these solutions. If you made a mistake, don’t worry! Look at the explanation to understand where you went wrong.

  1. Solution:

    • Distribute: $2a^2b * 3ab^2 = 6a3b3$
    • Distribute: $2a^2b * -5a^2b = -10a4b2$
    • Simplified: $6a3b3 - 10a4b2$

  2. Solution:

    • Distribute: $-4pq^3 * 2p^2q = -8p3q4$
    • Distribute: $-4pq^3 * -6pq^2 = 24p2q5$
    • Simplified: $-8p3q4 + 24p2q5$

By practicing and reviewing, you’ll build a solid foundation in simplifying algebraic expressions. Keep practicing, and you'll get the hang of it!