Simplify The Expression: 2j(2j² + 5)

Hey guys! Let's dive into simplifying this mathematical expression. We've got $2j(2j^2 + 5)$, and our goal is to make it as neat and tidy as possible. Think of it like decluttering a room – we want to organize the terms and get rid of any unnecessary complexity. So, grab your mathematical tool belts, and let's get started!

Breaking Down the Expression

First, let's understand what we're dealing with. The expression $2j(2j^2 + 5)$ involves a term outside the parentheses ($2j$) and a binomial (two-term expression) inside the parentheses ($2j^2 + 5$). To simplify this, we'll use the distributive property. This property is a fundamental concept in algebra, and it's super handy for this kind of problem. It basically says that we need to multiply the term outside the parentheses by each term inside the parentheses.

The distributive property can be expressed as:
$a(b + c) = ab + ac$

In our case, $a$ is $2j$, $b$ is $2j^2$, and $c$ is $5$. So, we're going to multiply $2j$ by both $2j^2$ and $5$. This step is crucial because it allows us to eliminate the parentheses and combine like terms later on. It's like unlocking a door that leads to a simpler form of the expression.

Applying the Distributive Property

Now, let's apply the distributive property step-by-step:

1. Multiply $2j$ by $2j^2$:
   $(2j) * (2j^2) = 4j^3$
   Here, we multiply the coefficients (2 * 2 = 4) and add the exponents of $j$ (j^1 * j^2 = j^(1+2) = j^3$). Remember, when multiplying terms with exponents, we add the exponents if the bases are the same. This is a key rule in algebra that helps us keep our calculations accurate.
2. Multiply $2j$ by $5$:
   $(2j) * (5) = 10j$
   This is a straightforward multiplication. We simply multiply the coefficient 2 by 5, and keep the variable $j$. So, we get 10j.

Combining the Results

After applying the distributive property, we have two terms: $4j^3$ and $10j$. Now we add these terms together to get our expanded expression:

$4j^3 + 10j$

This is our simplified expression! We've successfully distributed the $2j$ across the terms inside the parentheses, resulting in a clearer and more manageable form.

Final Simplified Expression

So, after applying the distributive property and combining the terms, we arrive at our simplified expression:

$4j^3 + 10j$

This is the final answer. We've taken the original expression, $2j(2j^2 + 5)$, and simplified it to $4j^3 + 10j$. There are no more like terms to combine, and the expression is now in its simplest form. High five! You've successfully navigated through this algebraic simplification. Keep up the great work, and remember, practice makes perfect!

Why is this simplification important?

You might be wondering, why bother simplifying expressions in the first place? Well, simplifying expressions makes them easier to work with in further calculations. Imagine trying to solve an equation or graph a function with a complicated expression – it would be a nightmare! Simplified expressions are like a clean, organized workspace; they help prevent errors and make complex problems more manageable. It's like having a clear roadmap instead of a tangled mess of streets when you're trying to find your way.

Moreover, simplification is crucial in various fields such as physics, engineering, and computer science. In physics, simplified equations can help describe the motion of objects or the behavior of circuits. In engineering, they can aid in designing structures or systems. In computer science, they can optimize algorithms and improve performance. So, mastering simplification techniques is not just an academic exercise; it's a valuable skill that can be applied in countless real-world scenarios.

Common Mistakes to Avoid

When simplifying expressions, there are a few common pitfalls to watch out for. One of the most frequent errors is incorrectly applying the distributive property. Remember, you need to multiply the term outside the parentheses by every term inside the parentheses. Forgetting to multiply by one of the terms can lead to an incorrect simplification. It's like baking a cake and forgetting an ingredient – the result won't be quite right.

Another mistake is incorrectly adding exponents when multiplying terms. Remember, when you multiply terms with the same base, you add the exponents, not multiply them. For instance, $j^2 * j^3$ is $j^5$, not $j^6$. Getting this rule wrong can throw off your entire calculation. It’s similar to misreading a musical note – it can disrupt the harmony of the entire piece.

Finally, be careful when combining like terms. Only terms with the same variable and exponent can be combined. For example, $4j^3$ and $10j$ cannot be combined because they have different exponents. Trying to combine unlike terms is like trying to mix oil and water – they just don't blend. Always double-check that you're only combining terms that are truly alike.

Practice Problems

To solidify your understanding, let's try a few more practice problems. Remember, the key to mastering simplification is practice, practice, practice! It's like learning a new language – the more you use it, the more fluent you become.

Problem 1

Simplify: $3k(4k^2 - 2)$

Solution

1. Apply the distributive property:
   $(3k) * (4k^2) = 12k^3$
   $(3k) * (-2) = -6k$
2. Combine the terms:
   $12k^3 - 6k$

Problem 2

Simplify: $-2m(5m^2 + 3m)$

Solution

1. Apply the distributive property:
   $(-2m) * (5m^2) = -10m^3$
   $(-2m) * (3m) = -6m^2$
2. Combine the terms:
   $-10m^3 - 6m^2$

Problem 3

Simplify: $4p(2p^2 - 7p + 1)$

Solution

1. Apply the distributive property:
   $(4p) * (2p^2) = 8p^3$
   $(4p) * (-7p) = -28p^2$
   $(4p) * (1) = 4p$
2. Combine the terms:
   $8p^3 - 28p^2 + 4p$

Conclusion

Simplifying expressions is a fundamental skill in algebra that opens the door to solving more complex problems. By understanding and applying the distributive property, you can break down expressions into simpler, more manageable forms. Remember to distribute carefully, combine like terms accurately, and avoid common mistakes. With practice, you'll become a simplification superstar! Keep honing your skills, and you'll be well-equipped to tackle any algebraic challenge that comes your way. You've got this!