Simplify $47a^2 - (-8a^2)$: A Step-by-Step Guide

Hey guys! Let's dive into simplifying algebraic expressions. In this article, we'll break down how to simplify the expression $47a^2 - (-8a^2)$. This is a common type of problem you might encounter in algebra, and understanding how to solve it will definitely boost your math skills. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into the problem, it's essential to understand the basic principles involved. In algebra, we often deal with variables (like 'a' in our expression) and coefficients (the numbers in front of the variables). The key here is to remember how to handle negative signs and combine like terms.

Like terms are terms that have the same variable raised to the same power. In our case, $47a^2$ and $-8a^2$ are like terms because they both have 'a' raised to the power of 2. This allows us to combine them easily.

When you see a negative sign in front of a parenthesis, it means you're subtracting everything inside the parenthesis. Subtracting a negative number is the same as adding its positive counterpart. This is a crucial concept to keep in mind as we tackle the problem.

Now that we've refreshed our memory on these basics, let's move on to the step-by-step solution.

Step-by-Step Solution

Our expression is $47a^2 - (-8a^2)$. The first thing we need to address is the subtraction of the negative term. Remember, subtracting a negative is the same as adding a positive. So, we can rewrite the expression as:

$47a^2 + 8a^2$

Now that we've eliminated the double negative, we can combine the like terms. To do this, we simply add the coefficients:

$47 + 8 = 55$

So, the simplified expression is:

$55a^2$

And that's it! We've successfully simplified the expression $47a^2 - (-8a^2)$ to $55a^2$. This process involves understanding the rules of subtracting negative numbers and combining like terms.

Why This Matters

Understanding how to simplify algebraic expressions is crucial for several reasons. First, it's a fundamental skill in algebra and higher-level mathematics. You'll encounter similar problems in various contexts, from solving equations to working with polynomials.

Second, it helps develop your problem-solving skills. Breaking down a complex expression into simpler terms teaches you how to approach problems methodically and logically. This skill is transferable to many other areas of life.

Third, it builds a solid foundation for more advanced topics in mathematics. As you progress in your math education, you'll encounter more complex expressions and equations. Having a strong grasp of basic simplification techniques will make these topics much easier to understand.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. $25b^2 - (-12b^2)$
2. $-15c^2 - (-30c^2)$
3. $10x^2 - (-5x^2) + 3x^2$

Try solving these problems on your own, and then check your answers. The key is to remember the rules of subtracting negative numbers and combining like terms.

Common Mistakes to Avoid

When simplifying expressions like $47a^2 - (-8a^2)$, there are a few common mistakes you should watch out for. One of the most common is forgetting to correctly handle the negative signs.

Incorrectly Handling Negative Signs: A frequent error is not recognizing that subtracting a negative number is the same as adding a positive number. For example, some might mistakenly calculate $47a^2 - (-8a^2)$ as $47a^2 - 8a^2$, which would lead to an incorrect answer.

Combining Unlike Terms: Another mistake is attempting to combine terms that are not like terms. Remember, like terms must have the same variable raised to the same power. For example, you cannot combine $47a^2$ with $47a^3$ because the powers of 'a' are different.

Arithmetic Errors: Simple arithmetic errors can also lead to incorrect answers. Always double-check your calculations to ensure accuracy.

By being aware of these common mistakes, you can avoid them and ensure you're simplifying expressions correctly.

Real-World Applications

You might be wondering, where does this stuff actually get used in the real world? Well, simplifying algebraic expressions is a fundamental skill that comes in handy in various fields.

Engineering: Engineers use algebraic expressions to model and analyze systems. Simplifying these expressions helps them make accurate predictions and design efficient solutions.

Physics: Physicists use algebraic expressions to describe the laws of nature. Simplifying these expressions is crucial for solving problems and making calculations.

Computer Science: Computer scientists use algebraic expressions to develop algorithms and write code. Simplifying these expressions can help optimize code and improve performance.

Economics: Economists use algebraic expressions to model economic systems. Simplifying these expressions can help them understand economic trends and make predictions.

Finance: Financial analysts use algebraic expressions to analyze investments and manage risk. Simplifying these expressions can help them make informed decisions.

As you can see, simplifying algebraic expressions is a valuable skill that has applications in many different fields. By mastering this skill, you'll be well-equipped to tackle a wide range of problems.

Conclusion

Alright, guys, we've covered a lot in this article! We've learned how to simplify the expression $47a^2 - (-8a^2)$ by understanding the basics of algebra, handling negative signs, and combining like terms. We've also discussed why this skill is important and how it's used in the real world.

Remember, practice makes perfect. So, keep practicing with different expressions and don't be afraid to make mistakes. Mistakes are a natural part of the learning process, and they can actually help you learn more effectively.

So, go out there and simplify some expressions! And remember, math can be fun if you approach it with the right attitude. Keep up the great work, and I'll see you in the next article!