Simplify Algebraic Expression -5a^2-[3a-(-18a^2-9a)]-[-4+(-4a+4)]

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It involves the process of eliminating grouping symbols such as parentheses, brackets, and braces, and then combining like terms to arrive at a more concise and manageable form of the expression. This skill is crucial for solving equations, evaluating formulas, and tackling more advanced mathematical concepts.

In this comprehensive guide, we will delve into the intricacies of simplifying expressions, providing a step-by-step approach that will empower you to confidently tackle any algebraic expression. We will explore the order of operations, the distributive property, and the process of combining like terms, equipping you with the tools to master this essential mathematical technique.

Understanding the Order of Operations

Before we embark on the journey of simplifying expressions, it's essential to grasp 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 to ensure consistent and accurate results.

  1. Parentheses: The operations within parentheses, brackets, or braces are always performed first. This includes any mathematical operations enclosed within these grouping symbols.
  2. Exponents: Next, we evaluate exponents or powers. This involves raising a base number to a specified power, indicating how many times the base number is multiplied by itself.
  3. Multiplication and Division: Multiplication and division are performed from left to right. This means that if both operations appear in an expression, we perform them in the order they appear, moving from left to right.
  4. Addition and Subtraction: Finally, addition and subtraction are performed from left to right, similar to multiplication and division.

Adhering to the order of operations is paramount when simplifying expressions. Failing to do so can lead to incorrect results and a misunderstanding of the expression's true value. Let's illustrate this with an example:

Consider the expression 2 + 3 * 4. If we were to perform addition before multiplication, we would arrive at 5 * 4 = 20. However, following the order of operations, we perform multiplication first: 3 * 4 = 12, and then add 2 to get the correct answer, 14.

The Distributive Property: Unveiling Hidden Multiplications

The distributive property is a powerful tool in simplifying expressions, especially when dealing with expressions enclosed in parentheses. It states that multiplying a number by a sum or difference is the same as multiplying the number by each term inside the parentheses individually and then adding or subtracting the results.

Mathematically, the distributive property can be expressed as:

  • a(b + c) = ab + ac
  • a(b - c) = ab - ac

Where a, b, and c represent any numbers or variables.

Let's illustrate the distributive property with an example:

Consider the expression 3(x + 2). To simplify this expression, we distribute the 3 to both terms inside the parentheses:

  • 3(x + 2) = 3 * x + 3 * 2
  • = 3x + 6

The distributive property is particularly useful when dealing with expressions that involve variables or multiple terms within parentheses. It allows us to remove the parentheses and combine like terms, leading to a simplified expression.

Combining Like Terms: Uniting the Similar

Like terms are terms that have the same variable raised to the same power. For example, 3x and 5x are like terms because they both have the variable x raised to the power of 1. Similarly, 2y^2 and -7y^2 are like terms because they both have the variable y raised to the power of 2. However, 4x and 4x^2 are not like terms because the variable x is raised to different powers.

Combining like terms involves adding or subtracting their coefficients, which are the numerical factors that multiply the variables. To combine like terms, we simply add or subtract their coefficients while keeping the variable and its exponent the same.

Let's illustrate this with an example:

Consider the expression 2x + 3y - 5x + 2y. To combine like terms, we identify the terms with the same variable and exponent:

  • 2x and -5x are like terms.
  • 3y and 2y are like terms.

Now, we combine the like terms by adding or subtracting their coefficients:

  • 2x - 5x = -3x
  • 3y + 2y = 5y

Therefore, the simplified expression is -3x + 5y.

Combining like terms is a crucial step in simplifying expressions. It allows us to reduce the number of terms in an expression and make it easier to work with.

Step-by-Step Guide to Simplifying Expressions

Now that we have explored the order of operations, the distributive property, and the process of combining like terms, let's outline a step-by-step guide to simplifying expressions:

  1. Address Grouping Symbols: Begin by simplifying any expressions within parentheses, brackets, or braces, working from the innermost grouping symbols outwards. Remember to adhere to the order of operations within each set of grouping symbols.
  2. Apply the Distributive Property: If there are any terms multiplied by expressions within parentheses, use the distributive property to remove the parentheses.
  3. Identify Like Terms: Identify terms that have the same variable raised to the same power.
  4. Combine Like Terms: Combine like terms by adding or subtracting their coefficients.
  5. Arrange Terms in Descending Order of Powers: Arrange the terms in the simplified expression in descending order of their exponents. This is a standard practice that helps to present the expression in a clear and organized manner.

Example: Simplifying a Complex Expression

Let's put our knowledge into practice by simplifying a complex expression:

Simplify the expression -5a^2 - [3a - (-18a^2 - 9a)] - [-4 + (-4a + 4)]

  1. Address Grouping Symbols: We begin by simplifying the expressions within the innermost grouping symbols, which are the parentheses:
    • -(-18a^2 - 9a) = 18a^2 + 9a
    • (-4a + 4) = -4a + 4

The expression now becomes: -5a^2 - [3a + 18a^2 + 9a] - [-4 - 4a + 4] 2. Address Grouping Symbols: Next, we simplify the expressions within the brackets: * [3a + 18a^2 + 9a] = 18a^2 + 12a * [-4 - 4a + 4] = -4a

The expression now becomes: -5a^2 - (18a^2 + 12a) - (-4a) 3. Apply the Distributive Property: We distribute the negative signs in front of the parentheses: * -(18a^2 + 12a) = -18a^2 - 12a * -(-4a) = 4a

The expression now becomes: -5a^2 - 18a^2 - 12a + 4a 4. Identify Like Terms: We identify the like terms: * -5a^2 and -18a^2 are like terms. * -12a and 4a are like terms. 5. Combine Like Terms: We combine the like terms: * -5a^2 - 18a^2 = -23a^2 * -12a + 4a = -8a

The expression now becomes: -23a^2 - 8a 6. Arrange Terms in Descending Order of Powers: The terms are already arranged in descending order of powers.

Therefore, the simplified expression is -23a^2 - 8a.

Common Mistakes to Avoid

Simplifying expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

  • Forgetting the Order of Operations: Always adhere to the order of operations (PEMDAS) to ensure accurate results.
  • Incorrectly Distributing: When applying the distributive property, make sure to multiply the term outside the parentheses by each term inside the parentheses, paying attention to signs.
  • Combining Unlike Terms: Only combine terms that have the same variable raised to the same power.
  • Sign Errors: Pay close attention to signs when combining like terms and distributing negative signs.

By avoiding these common mistakes, you can significantly improve your accuracy and confidence in simplifying expressions.

Conclusion: Mastering the Art of Simplification

Simplifying expressions is a fundamental skill in mathematics that forms the basis for more advanced concepts. By understanding the order of operations, the distributive property, and the process of combining like terms, you can confidently tackle any algebraic expression. Remember to follow the step-by-step guide, avoid common mistakes, and practice regularly to hone your skills.

With dedication and practice, you can master the art of simplifying expressions and unlock a deeper understanding of the mathematical world. This skill will not only serve you well in mathematics courses but also in various real-world applications where algebraic manipulation is required.

To simplify the expression −5a2−[3a−(−18a2−9a)]−[−4+(−4a+4)]-5a^2-[3a-(-18a^2-9a)]-[-4+(-4a+4)], we will follow the order of operations and combine like terms step by step. This process involves removing the grouping symbols and then combining the terms that have the same variable and exponent.

Step-by-Step Solution

Here is a detailed breakdown of how to simplify the given expression:

  1. Start with the innermost parentheses: The expression inside the innermost parentheses is −18a2−9a-18a^2 - 9a. The negative sign in front of this parenthesis changes the signs of the terms inside when we remove it.

    • −5a2−[3a−(−18a2−9a)]−[−4+(−4a+4)]-5a^2 - [3a - (-18a^2 - 9a)] - [-4 + (-4a + 4)] becomes −5a2−[3a+18a2+9a]−[−4+(−4a+4)]-5a^2 - [3a + 18a^2 + 9a] - [-4 + (-4a + 4)]

  2. Simplify inside the first bracket: Combine like terms inside the first bracket [3a+18a2+9a][3a + 18a^2 + 9a].

    • 3a+9a3a + 9a are like terms, so we add them: 3a+9a=12a3a + 9a = 12a
    • The expression inside the first bracket simplifies to 18a2+12a18a^2 + 12a
    • Now the expression looks like: −5a2−[18a2+12a]−[−4+(−4a+4)]-5a^2 - [18a^2 + 12a] - [-4 + (-4a + 4)]

  3. Simplify inside the second bracket: In the second bracket [−4+(−4a+4)][-4 + (-4a + 4)], we have constants and a term with aa.

    • First, combine the constants: −4+4=0-4 + 4 = 0
    • The bracket simplifies to −4a-4a
    • Now the entire expression is: −5a2−[18a2+12a]−[−4a]-5a^2 - [18a^2 + 12a] - [-4a]

  4. Remove the brackets: Distribute the negative signs in front of the brackets.

    • −[18a2+12a]- [18a^2 + 12a] becomes −18a2−12a-18a^2 - 12a
    • −[−4a]- [-4a] becomes +4a+4a
    • The expression now looks like: −5a2−18a2−12a+4a-5a^2 - 18a^2 - 12a + 4a

  5. Combine like terms: Now, we combine like terms in the expression −5a2−18a2−12a+4a-5a^2 - 18a^2 - 12a + 4a.

    • Combine the a2a^2 terms: −5a2−18a2=−23a2-5a^2 - 18a^2 = -23a^2
    • Combine the aa terms: −12a+4a=−8a-12a + 4a = -8a

  6. Write the simplified expression: The simplified expression is −23a2−8a-23a^2 - 8a.

Final Answer

The simplified expression, with terms written from the highest to the lowest power of the variable, is: −23a2−8a-23a^2 - 8a.

This step-by-step solution ensures that the expression is simplified correctly by following the order of operations and combining like terms methodically. Understanding and practicing these simplifications are crucial in algebra for solving more complex equations and problems.