Simplify -15a-{12a+[3a-(-5a-13)]} Remove Grouping Symbols And Combine Like Terms

In mathematics, simplifying algebraic expressions is a fundamental skill. It involves removing grouping symbols like parentheses, brackets, and braces, and then combining like terms to express the expression in its most concise form. This not only makes the expression easier to understand but also facilitates further algebraic manipulations. In this comprehensive guide, we will walk through the process of simplifying the expression $-15a - \{12a + [3a - (-5a - 13)]\}$ step-by-step, ensuring clarity and understanding at each stage.

Understanding the Order of Operations and Grouping Symbols

Before diving into the simplification process, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed. Grouping symbols, such as parentheses (), brackets [], and braces {}, are used to indicate which operations should be performed first. When dealing with nested grouping symbols, we work from the innermost to the outermost.

Grouping symbols play a vital role in algebra, as they dictate the order in which operations are performed. Understanding how to handle these symbols is essential for simplifying expressions correctly. We always start with the innermost grouping symbols and work our way outwards. This ensures that we follow the correct order of operations and arrive at the correct simplified expression. Ignoring the correct order can lead to errors and an incorrect final result. The expression we aim to simplify, $-15a - \{12a + [3a - (-5a - 13)]\}$, involves nested grouping symbols: parentheses, brackets, and braces. This nesting requires a systematic approach, starting with the innermost parentheses and progressing outwards. Mastering the handling of grouping symbols not only simplifies complex expressions but also lays a solid foundation for more advanced algebraic concepts. As we proceed, we will demonstrate how to carefully remove each layer of grouping symbols while maintaining the integrity of the expression. Remember, the key is to take it one step at a time, ensuring that each operation is performed in the correct sequence.

Step 1: Removing the Innermost Parentheses

Our first step is to tackle the innermost grouping symbol, which is the parentheses in the term $(-5a - 13)$. To remove these parentheses, we need to distribute the negative sign that precedes them. This means multiplying each term inside the parentheses by -1. This is a crucial step because incorrect distribution of the negative sign is a common source of errors in algebraic simplification.

Distributing the negative sign in $-(-5a - 13)$ requires careful attention to detail. When we multiply $-5a$ by -1, we get $+5a$. Similarly, when we multiply -13 by -1, we get +13. Thus, $-(-5a - 13)$ simplifies to $5a + 13$. This step is a fundamental application of the distributive property, which states that $a(b + c) = ab + ac$. In this case, we are distributing -1 across the terms inside the parentheses. It's essential to remember that multiplying a negative by a negative results in a positive. This simple rule is the cornerstone of this step and must be applied accurately to avoid errors. Once we have correctly distributed the negative sign, the expression becomes considerably simpler, paving the way for the next steps in the simplification process. The ability to handle negative signs correctly is a key skill in algebra, and this step provides a clear example of its importance. By mastering this technique, you will be better equipped to tackle more complex algebraic manipulations.

Now, let's rewrite the expression with the parentheses removed:

$-15a - \{12a + [3a + 5a + 13]\}$

Step 2: Simplifying Within the Brackets

Now that we've eliminated the parentheses, our attention turns to the brackets. Inside the brackets, we have the expression $[3a + 5a + 13]$. The next logical step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, $3a$ and $5a$ are like terms, as they both have the variable 'a' raised to the power of 1. Combining like terms is a fundamental simplification technique in algebra.

Combining $3a$ and $5a$ involves simply adding their coefficients. The coefficient is the numerical part of the term. So, we add the coefficients 3 and 5, which gives us 8. Thus, $3a + 5a$ simplifies to $8a$. The expression inside the brackets now becomes $8a + 13$. This step demonstrates the power of combining like terms to reduce the complexity of an expression. By grouping terms that share the same variable and exponent, we make the expression more manageable and easier to work with. This skill is crucial for simplifying more complex algebraic expressions and solving equations. It is also an essential part of building a strong foundation in algebra. The ability to quickly and accurately combine like terms will save time and reduce the likelihood of errors in subsequent steps. So, remember to always look for like terms when simplifying algebraic expressions and combine them to make the expression simpler.

Substituting this back into the expression, we get:

$-15a - \{12a + [8a + 13]\}$

Next, we remove the brackets:

$-15a - \{12a + 8a + 13\}$

Step 3: Simplifying Within the Braces

We now move on to the outermost grouping symbols, the braces. Inside the braces, we have the expression $\{12a + 8a + 13\}$. Similar to the previous step, we need to combine like terms. Here, $12a$ and $8a$ are like terms, as they both contain the variable 'a' raised to the power of 1. Combining these terms will further simplify the expression within the braces.

Combining the like terms $12a$ and $8a$ is a straightforward addition of their coefficients. We add 12 and 8, which equals 20. Therefore, $12a + 8a$ simplifies to $20a$. The expression within the braces now becomes $20a + 13$. This step is a continuation of the process of making the expression more concise and manageable. It reinforces the importance of recognizing and combining like terms to simplify algebraic expressions. This technique is not only useful for simplification but also crucial for solving equations and other algebraic problems. The ability to quickly identify and combine like terms is a valuable skill that will serve you well in various mathematical contexts. So, always be on the lookout for like terms when working with algebraic expressions and combine them to make your work easier and more efficient.

Rewriting the expression, we have:

$-15a - \{20a + 13\}$

Now, we remove the braces. Since there is a negative sign in front of the braces, we need to distribute the negative sign to each term inside the braces:

$-15a - 20a - 13$

Step 4: Combining Remaining Like Terms

After removing all grouping symbols, we are left with the expression $-15a - 20a - 13$. Our final step is to combine the remaining like terms. In this expression, $-15a$ and $-20a$ are like terms. Combining these terms will give us the simplest form of the original expression.

To combine $-15a$ and $-20a$, we add their coefficients: -15 + (-20) = -35. Therefore, $-15a - 20a$ simplifies to $-35a$. The expression now becomes $-35a - 13$. This final step completes the simplification process, resulting in a concise and easily understandable expression. Combining like terms is a fundamental technique in algebra, allowing us to reduce complex expressions to their simplest forms. It is a skill that is used extensively in various mathematical contexts, from solving equations to simplifying complex formulas. Mastering this technique is essential for success in algebra and beyond. So, always remember to look for like terms and combine them to simplify expressions and make your work more efficient.

The simplified expression is:

$-35a - 13$

Step 5: Writing Terms in Descending Order of Power

The final step in simplifying algebraic expressions often involves arranging the terms in descending order of their powers. This means that the term with the highest power of the variable comes first, followed by terms with lower powers, and finally the constant term. In our simplified expression, $-35a - 13$, the term $-35a$ has 'a' raised to the power of 1, and -13 is a constant term (which can be thought of as having 'a' raised to the power of 0).

In the expression $-35a - 13$, the term with the highest power of the variable 'a' is $-35a$, which has 'a' raised to the power of 1. The term -13 is a constant term, which can be considered as having 'a' raised to the power of 0. Therefore, the expression is already in the correct order, with the term containing the highest power of the variable appearing first, followed by the constant term. This arrangement is a standard practice in algebra, as it makes expressions easier to read and interpret. It also facilitates further algebraic manipulations, such as factoring and solving equations. So, when simplifying algebraic expressions, remember to arrange the terms in descending order of their powers to ensure clarity and consistency.

In this case, the expression is already in the desired order:

$-35a - 13$

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By following the order of operations, systematically removing grouping symbols, and combining like terms, we can transform complex expressions into simpler, more manageable forms. This step-by-step guide has demonstrated the process using the example $-15a - \{12a + [3a - (-5a - 13)]\}$, resulting in the simplified expression $-35a - 13$. Remember to always work from the innermost grouping symbols outwards and pay close attention to signs when distributing. With practice, you'll become proficient in simplifying algebraic expressions, paving the way for success in more advanced mathematical concepts.

Mastering these techniques is crucial for success in algebra and beyond. By understanding the order of operations, handling grouping symbols correctly, and combining like terms effectively, you can confidently tackle a wide range of algebraic simplification problems. Remember, practice makes perfect, so continue to apply these techniques to various expressions to solidify your understanding and skills.