Calculate M+n For Similar Monomials And Expression Reduction

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In the realm of algebra, monomials hold a fundamental position as single-term expressions, often involving variables raised to various powers. The concept of similar monomials arises when these expressions share the same variables raised to the same exponents, allowing for algebraic manipulation and simplification. This article delves into the process of calculating the sum of variables, specifically 'm' and 'n,' when presented with similar monomials P and Q. We will explore the underlying principles, demonstrate the solution step-by-step, and extend the discussion to encompass the reduction, addition, and subtraction of algebraic expressions.

Understanding Similar Monomials

At the heart of this problem lies the concept of similar monomials. Two monomials are deemed similar if they possess the same variables raised to the same powers. This seemingly simple criterion forms the bedrock for algebraic manipulations, allowing us to combine like terms and simplify complex expressions. To illustrate, consider the monomials 3x²y and -5x²y. These are similar because they both contain the variables 'x' and 'y' raised to the powers of 2 and 1, respectively. On the other hand, 2xy² and 4x²y are not similar due to the differing exponents of 'x' and 'y'. Recognizing similar monomials is crucial for performing addition, subtraction, and other algebraic operations.

Problem Statement

We are presented with two monomials, P(x; y) and Q(x; y), defined as follows:

P(x; y) = 8x^(m-1)y^8
Q(x; y) = -5x^7y^(2n+2)

The problem states that P and Q are similar monomials. Our objective is to calculate the sum of 'm' and 'n'.

Solution

The key to solving this problem lies in the definition of similar monomials. Since P and Q are similar, their variables must have the same exponents. This allows us to set up a system of equations based on the exponents of 'x' and 'y'.

Equating Exponents

For the variable 'x', the exponent in P is (m-1), and the exponent in Q is 7. Therefore, we have the equation:

m - 1 = 7

Solving for 'm', we add 1 to both sides:

m = 7 + 1
m = 8

For the variable 'y', the exponent in P is 8, and the exponent in Q is (2n+2). This gives us the equation:

8 = 2n + 2

Solving for 'n', we first subtract 2 from both sides:

6 = 2n

Then, we divide both sides by 2:

n = 3

Calculating m + n

Now that we have determined the values of 'm' and 'n', we can calculate their sum:

m + n = 8 + 3
m + n = 11

Therefore, the sum of 'm' and 'n' is 11.

Reduce, Apply Addition, and Subtraction of Expressions

Beyond calculating the sum of variables in similar monomials, the principles of algebraic manipulation extend to reducing expressions and applying addition and subtraction. This section explores these concepts, providing a comprehensive understanding of how to simplify and combine algebraic expressions.

Reducing Expressions

Reducing an algebraic expression involves simplifying it by combining like terms and eliminating unnecessary elements. This process often makes the expression easier to understand and work with. The key to reducing expressions lies in identifying and combining like terms. Like terms are terms that have the same variables raised to the same powers. For instance, 3x²y and -5x²y are like terms, while 2xy² and 4x²y are not.

To reduce an expression, we follow these steps:

  1. Identify like terms: Group terms that have the same variables raised to the same powers.
  2. Combine like terms: Add or subtract the coefficients of like terms. The variables and their exponents remain the same.

For example, consider the expression:

5x² + 3xy - 2x² + 7xy - x²

First, we identify the like terms:

  • 5x², -2x², and -x² are like terms.
  • 3xy and 7xy are like terms.

Next, we combine the like terms:

(5x² - 2x² - x²) + (3xy + 7xy)
= (5 - 2 - 1)x² + (3 + 7)xy
= 2x² + 10xy

Therefore, the reduced expression is 2x² + 10xy. This process of simplification is crucial in various mathematical and scientific contexts.

Addition of Expressions

Adding algebraic expressions involves combining two or more expressions into a single expression. This process is similar to reducing expressions, as we combine like terms. To add expressions, we follow these steps:

  1. Write the expressions: Write the expressions to be added, separated by a plus sign (+).
  2. Identify like terms: Group like terms from all the expressions.
  3. Combine like terms: Add the coefficients of like terms. The variables and their exponents remain the same.

For example, consider adding the expressions (3a + 2b - c) and (5a - 4b + 3c):

  1. Write the expressions:
(3a + 2b - c) + (5a - 4b + 3c)
  1. Identify like terms:
  • 3a and 5a are like terms.
  • 2b and -4b are like terms.
  • -c and 3c are like terms.
  1. Combine like terms:
(3a + 5a) + (2b - 4b) + (-c + 3c)
= 8a - 2b + 2c

Therefore, the sum of the expressions is 8a - 2b + 2c. Understanding the addition of expressions is fundamental in various algebraic manipulations.

Subtraction of Expressions

Subtracting algebraic expressions involves finding the difference between two or more expressions. This process is similar to addition, but we subtract the coefficients of like terms instead of adding them. To subtract expressions, we follow these steps:

  1. Write the expressions: Write the expressions to be subtracted, separated by a minus sign (-). It's often helpful to enclose the expression being subtracted in parentheses.
  2. Distribute the negative sign: Distribute the negative sign to each term inside the parentheses of the expression being subtracted. This means changing the sign of each term.
  3. Identify like terms: Group like terms from all the expressions.
  4. Combine like terms: Add or subtract the coefficients of like terms. The variables and their exponents remain the same.

For example, consider subtracting the expression (2x² - 3xy + y²) from (5x² + xy - 2y²):

  1. Write the expressions:
(5x² + xy - 2y²) - (2x² - 3xy + y²)
  1. Distribute the negative sign:
5x² + xy - 2y² - 2x² + 3xy - y²
  1. Identify like terms:
  • 5x² and -2x² are like terms.
  • xy and 3xy are like terms.
  • -2y² and -y² are like terms.
  1. Combine like terms:
(5x² - 2x²) + (xy + 3xy) + (-2y² - y²)
= 3x² + 4xy - 3y²

Therefore, the difference between the expressions is 3x² + 4xy - 3y². Mastering the subtraction of expressions is essential for solving algebraic equations and simplifying complex expressions.

Conclusion

In conclusion, determining the sum of variables 'm' and 'n' in similar monomials involves equating the exponents of corresponding variables and solving the resulting system of equations. This process underscores the fundamental concept of similar monomials and their role in algebraic manipulation. Furthermore, the principles of reducing expressions, adding, and subtracting algebraic expressions provide a comprehensive toolkit for simplifying and combining algebraic expressions. These skills are crucial for success in algebra and other areas of mathematics. By understanding and applying these concepts, students and practitioners can confidently tackle a wide range of algebraic problems and gain a deeper appreciation for the elegance and power of algebraic methods. Practice and application are key to mastering these concepts and building a strong foundation in algebra.