Simplifying Algebraic Expressions: -13z³ - 44z³

Hey guys! Let's dive into simplifying the expression $-13z^3 - 44z^3$. This might look a bit intimidating at first, but trust me, it's super manageable once we break it down. We are essentially dealing with like terms here, and combining like terms is a fundamental concept in algebra. It's like adding apples and apples – you just count how many you have in total. So, let's get started and make this algebraic expression a piece of cake!

Understanding Like Terms

Before we jump into the simplification, let's quickly recap what like terms actually are. Like terms are terms that have the same variable raised to the same power. Think of it this way: the variable is the letter (like z in our case), and the power is the little number that sits on top (like the 3 in $z^3$). So, for terms to be considered "like," they need to have the exact same letter with the exact same exponent. For example, $5z^3$ and $-2z^3$ are like terms because they both have z raised to the power of 3. However, $5z^3$ and $5z^2$ are not like terms because the exponents are different, even though they both have the variable z. Similarly, $5z^3$ and $5y^3$ are also not like terms because the variables are different.

In our expression, $-13z^3$ and $-44z^3$, we can clearly see that both terms have the variable z raised to the power of 3. This makes them like terms, which means we can combine them! This is awesome because it simplifies the whole process. Imagine trying to add apples and oranges – it doesn't quite work, right? But adding apples and apples? That’s a breeze! Same idea here. We can only combine terms that are alike, and in this case, we've got a pair of z^3 terms ready to be simplified.

Recognizing like terms is crucial in algebra because it allows us to condense and simplify expressions, making them easier to work with. Without this understanding, we'd be stuck with long, messy expressions that are hard to manipulate. So, keep an eye out for those like terms – they're your friends in the world of algebra!

Combining Like Terms

Okay, now that we've confirmed that $-13z^3$ and $-44z^3$ are indeed like terms, let's get down to the nitty-gritty of combining them. Combining like terms is essentially just adding or subtracting their coefficients. Coefficients are the numbers that multiply the variable part of the term. In our case, the coefficient of $-13z^3$ is $-13$, and the coefficient of $-44z^3$ is $-44$. So, what we need to do is add these coefficients together.

Think of it like this: if you have $-13$ of something and you add $-44$ more of the same thing, how many do you have in total? Mathematically, this looks like: $-13 + (-44)$. Remember your rules for adding negative numbers! Adding a negative number is the same as subtracting its positive counterpart. So, $-13 + (-44)$ is the same as $-13 - 44$. Now, we're subtracting a larger number from a smaller number (or, more accurately, adding two negative numbers), so our result will be negative. The magnitude of the result will be the sum of the magnitudes of the two numbers. In other words, we add 13 and 44, which gives us 57, and since both numbers were negative, our final sum is $-57$.

So, $-13 + (-44) = -57$. This means that when we combine the like terms $-13z^3$ and $-44z^3$, we get $-57z^3$. We simply add the coefficients and keep the variable part ($z^3$) the same. It's like saying we had $-13$ "z cubed" things and then we got $-44$ more "z cubed" things, so now we have a total of $-57$ "z cubed" things. Easy peasy, right?

This process of combining like terms is super useful because it helps us to simplify complex expressions into more manageable forms. By adding or subtracting the coefficients of like terms, we can reduce the number of terms in an expression and make it easier to solve equations or perform other algebraic operations. So, mastering this skill is definitely a win in your algebra toolkit!

The Simplification Process

Alright, let's walk through the simplification process step-by-step to make sure we've got it nailed down. We started with the expression $-13z^3 - 44z^3$. The first thing we did was identify that the terms $-13z^3$ and $-44z^3$ are like terms. Remember, like terms have the same variable raised to the same power. In this case, both terms have the variable z raised to the power of 3, so they're definitely like terms.

Once we've identified the like terms, the next step is to combine them. To combine like terms, we focus on their coefficients. The coefficient of $-13z^3$ is $-13$, and the coefficient of $-44z^3$ is $-44$. We need to add these coefficients together: $-13 + (-44)$. As we discussed earlier, adding a negative number is the same as subtracting, so we can rewrite this as $-13 - 44$.

Now, we perform the addition (or subtraction, in this case). $-13 - 44$ equals $-57$. So, the combined coefficient is $-57$. Finally, we take this combined coefficient and multiply it by the variable part, which is $z^3$. This gives us our simplified term: $-57z^3$.

Therefore, the simplified form of the expression $-13z^3 - 44z^3$ is $-57z^3$. We've successfully reduced two terms into a single term, making the expression much simpler. This step-by-step process is a great way to tackle any simplification problem involving like terms. Just remember to identify the like terms, combine their coefficients, and then write the result with the appropriate variable part. Keep practicing, and you'll become a pro at simplifying algebraic expressions in no time!

Final Result

So, guys, we've successfully simplified the expression! After identifying the like terms, combining their coefficients, and putting it all together, we arrived at the final result: $-57z^3$. Isn't that satisfying? Taking a seemingly complex expression and boiling it down to its simplest form is what algebra is all about. We started with two terms, $-13z^3$ and $-44z^3$, and through the magic of combining like terms, we ended up with a single, neat term: $-57z^3$.

This result tells us that if you were to take $-13$ times $z^3$ and subtract another $44$ times $z^3$, you would end up with $-57$ times $z^3$. It's a concise way of representing the same mathematical relationship. And that's the power of simplification! It allows us to express things in the most straightforward and understandable manner.

I hope this breakdown has made the process clear and easy to follow. Remember, the key is to recognize like terms and then combine their coefficients. With a little practice, you'll be simplifying algebraic expressions like a champ. Keep up the great work, and don't hesitate to tackle more problems like this one. You've got this!