Simplifying Complex Algebraic Expressions Step-by-Step Guide

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In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article will guide you through the step-by-step simplification of the expression 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2. We will break down each term, apply algebraic principles, and combine like terms to arrive at the most simplified form. This process not only enhances your understanding of algebraic manipulation but also provides a practical approach to solving similar mathematical problems. Whether you are a student looking to improve your math skills or someone who enjoys the challenge of mathematical simplification, this article offers a comprehensive guide to tackling complex expressions with confidence.

Breaking Down the Expression

To effectively simplify the expression 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2, we must first understand each component and how they interact. The expression is composed of several terms, each requiring specific algebraic techniques for simplification. Let's dissect each term individually:

  1. 400: This is a constant term. Constant terms are numerical values that do not contain any variables. In the context of simplification, constants will be combined with other constant terms.
  2. (4y)^2: This term involves a variable, y, and an exponent. To simplify this, we need to apply the power rule, which states that (ab)^n = a^n * b^n. In this case, (4y)^2 becomes 4^2 * y^2, which simplifies to 16y^2. This is a quadratic term, meaning it involves the variable raised to the power of 2.
  3. 12(3m + 4y): This term involves the distributive property. The distributive property states that a(b + c) = ab + ac. We need to multiply 12 by each term inside the parentheses. So, 12(3m + 4y) becomes 12 * 3m + 12 * 4y, which simplifies to 36m + 48y. This introduces a term with the variable m and another term with the variable y.
  4. (600 + 5y): This is a linear expression. It contains a constant term (600) and a term with the variable y (5y). These terms are already in their simplest form individually, but they will need to be combined with like terms later in the simplification process.
  5. (200 + 4y)^2: This is the most complex term in the expression. It is a binomial (an expression with two terms) squared. To simplify this, we need to expand the binomial using the formula (a + b)^2 = a^2 + 2ab + b^2. Here, a = 200 and b = 4y. So, (200 + 4y)^2 becomes 200^2 + 2 * 200 * 4y + (4y)^2. This further simplifies to 40000 + 1600y + 16y^2. This term introduces a large constant, a linear term in y, and another quadratic term in y.

By breaking down the expression into these individual terms, we can systematically apply the appropriate algebraic rules and properties to simplify each part. This methodical approach is crucial for handling complex expressions and ensuring accuracy in the simplification process. In the following sections, we will delve into the specific steps for simplifying each term and combining like terms to reach the final simplified form.

Applying Algebraic Principles

After dissecting the expression 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2, the next crucial step is to apply algebraic principles to simplify each term. This involves using the order of operations (PEMDAS/BODMAS), the distributive property, and exponent rules. Let's go through each term one by one:

  1. Simplifying (4y)^2: As mentioned earlier, this term requires the application of the power rule. The power rule states that (ab)^n = a^n * b^n. Applying this rule to (4y)^2, we get 4^2 * y^2. Since 4^2 equals 16, the simplified form of this term is 16y^2.

  2. Simplifying 12(3m + 4y): This term involves the distributive property. The distributive property states that a(b + c) = ab + ac. Applying this property, we multiply 12 by each term inside the parentheses: 12 * 3m and 12 * 4y. This results in 36m + 48y. These two terms, 36m and 48y, cannot be combined further as they involve different variables.

  3. Simplifying (200 + 4y)^2: This is the most complex term and requires the use of the binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2. In this case, a = 200 and b = 4y. Applying the formula, we get:

    • a^2 = 200^2 = 40000
    • 2ab = 2 * 200 * 4y = 1600y
    • b^2 = (4y)^2 = 16y^2

    Combining these, we get 40000 + 1600y + 16y^2. This expanded form includes a constant, a linear term in y, and a quadratic term in y.

  4. The Terms 400 and (600 + 5y): The term 400 is already in its simplest form as it is a constant. The term (600 + 5y) is also in its simplest form as the constant and the term with y cannot be combined.

By applying these algebraic principles, we have successfully simplified each term in the original expression. The next step is to combine like terms. This involves identifying terms with the same variable and exponent and adding their coefficients. This process will further reduce the complexity of the expression and bring us closer to the final simplified form. In the following section, we will focus on combining these like terms to complete the simplification process.

Combining Like Terms

Having simplified each term in the expression 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2, the next critical step is to combine like terms. Like terms are terms that have the same variable raised to the same power. Combining like terms involves adding or subtracting their coefficients while keeping the variable and exponent the same. This process significantly reduces the complexity of the expression.

First, let's rewrite the expression with all the simplified terms:

400 + 16y^2 + 36m + 48y + 600 + 5y + 40000 + 1600y + 16y^2

Now, we will identify and combine the like terms:

  1. Constant Terms: We have the constants 400, 600, and 40000. Adding these together, we get:

    • 400 + 600 + 40000 = 41000

  2. Terms with y^2 (Quadratic Terms): We have 16y^2 and 16y^2. Combining these, we get:

    • 16y^2 + 16y^2 = 32y^2

  3. Terms with y (Linear Terms): We have 48y, 5y, and 1600y. Combining these, we get:

    • 48y + 5y + 1600y = 1653y

  4. Terms with m: We have only one term with m, which is 36m. Since there are no other like terms, it remains as 36m.

After combining all the like terms, the simplified expression is:

32y^2 + 1653y + 36m + 41000

This is the simplified form of the original expression. We have successfully reduced the complexity by combining like terms, making the expression easier to understand and work with. This process highlights the importance of recognizing like terms and applying the correct algebraic operations to simplify expressions. In the next section, we will discuss the final simplified expression and its implications.

Final Simplified Expression and Implications

After a detailed step-by-step simplification process, the expression 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2 has been reduced to its simplest form:

32y^2 + 1653y + 36m + 41000

This final expression is a polynomial containing terms with variables y and m, as well as a constant term. The expression is now in a more manageable and understandable format. Let's discuss the implications of this simplification and the characteristics of the final expression:

  1. Quadratic Term in y: The term 32y^2 is a quadratic term, indicating that the expression will exhibit parabolic behavior when graphed with respect to y. The coefficient 32 determines the direction and steepness of the parabola. Since the coefficient is positive, the parabola opens upwards.
  2. Linear Term in y: The term 1653y is a linear term, which contributes to the slope of the expression when graphed. The large coefficient of 1653 indicates a significant linear component in the overall behavior of the expression with respect to y.
  3. Linear Term in m: The term 36m is a linear term in m. This term is independent of y and will linearly affect the value of the expression based on the value of m. The coefficient 36 determines the rate of change of the expression with respect to m.
  4. Constant Term: The constant term 41000 represents the value of the expression when both y and m are zero. This term is significant as it sets the baseline value of the expression.
  5. Overall Behavior: The simplified expression allows for easier analysis of the expression's behavior. For instance, one can quickly determine how changes in y and m will affect the value of the expression. It also makes it easier to solve equations or inequalities involving the expression.

This simplification process underscores the importance of algebraic manipulation in making complex expressions more tractable. By breaking down the original expression, applying algebraic principles, and combining like terms, we have arrived at a simplified form that is both easier to interpret and use in further mathematical operations. This skill is essential in various fields, including engineering, physics, and computer science, where complex mathematical models often need to be simplified for analysis and computation.

In conclusion, the simplification of 400 + (4y)^2 + 12(3m + 4y) + (600 + 5y) + (200 + 4y)^2 to 32y^2 + 1653y + 36m + 41000 demonstrates the power of algebraic techniques in making complex mathematical expressions more accessible and understandable. This process not only simplifies the expression but also provides insights into its behavior and characteristics, making it a valuable skill in mathematical problem-solving.