Exercise 2.1 Solutions Summing Simple Monomials
This article provides detailed solutions and explanations for Exercise 2.1, which focuses on summing simple monomials. Monomials are algebraic expressions consisting of a single term, which can be a constant, a variable, or a product of constants and variables. Understanding how to add monomials is crucial for simplifying algebraic expressions and solving equations. This exercise covers various cases, including adding monomials with the same variables and exponents, as well as those with different terms. By working through these examples, you will strengthen your skills in basic algebraic manipulation and gain confidence in handling more complex expressions. Let's dive into the solutions and explore the underlying concepts.
Problem 1: 5b + 29b
Summing monomials often involves combining like terms. In this first problem, we are tasked with adding 5b and 29b. Both terms contain the variable b raised to the power of 1, making them like terms. To add them, we simply add their coefficients. The coefficients are the numerical parts of the terms, which are 5 and 29 in this case. Therefore, the sum is obtained by adding 5 and 29, resulting in 34. This means that 5b + 29b equals 34b. This basic operation highlights a fundamental principle in algebra: you can only directly add or subtract terms that have the same variable raised to the same power. Understanding this principle is essential for simplifying more complex algebraic expressions. For instance, trying to add 5b and 29b² directly would be incorrect because b and b² are different terms. The key is to identify the common variable and its exponent before proceeding with the addition. In this example, the straightforward addition of the coefficients provides a clear and concise solution, illustrating the core concept of combining like terms in monomials. The result, 34b, is a single term, maintaining the simplicity of a monomial. This lays the groundwork for handling more complex algebraic expressions where multiple terms need to be simplified and combined.
Solution:
5b + 29b = (5 + 29)b = 34b
Problem 2: (-68c) + 28c
This problem involves summing monomials with negative coefficients. Here, we need to add -68c and 28c. Like the previous problem, both terms contain the same variable, c, raised to the power of 1, making them like terms. To find the sum, we add the coefficients: -68 and 28. Adding these two numbers involves working with integers, where we have a negative number and a positive number. The result will have the sign of the number with the larger absolute value. In this case, the absolute value of -68 is 68, which is greater than the absolute value of 28. Therefore, the result will be negative. Subtracting 28 from 68 gives us 40. Hence, the sum of -68 and 28 is -40. Consequently, -68c + 28c equals -40c. This example highlights the importance of understanding how to work with negative numbers when summing monomials. It's a fundamental skill in algebra, ensuring accurate simplification of expressions. The ability to correctly handle negative coefficients is crucial, as it often appears in various algebraic problems. The solution -40c demonstrates the straightforward application of integer addition in the context of monomials.
Solution:
(-68c) + 28c = (-68 + 28)c = -40c
Problem 3: (-35m²) + (-78m²)
In this problem, we are summing monomials that involve the variable m raised to the power of 2. We need to add -35m² and -78m². Both terms are like terms because they have the same variable (m) and the same exponent (2). To add these terms, we add their coefficients: -35 and -78. When adding two negative numbers, we add their absolute values and keep the negative sign. The absolute value of -35 is 35, and the absolute value of -78 is 78. Adding 35 and 78 gives us 113. Since both numbers are negative, the sum will also be negative. Therefore, -35 + (-78) equals -113. This means that -35m² + (-78m²) simplifies to -113m². This example reinforces the concept of adding like terms and the rules for adding negative numbers. It is essential to remember that the exponent plays a crucial role in determining whether terms can be combined. Terms with different exponents, even if they have the same variable, cannot be added directly. This problem provides a clear illustration of how to handle negative coefficients and exponents when summing monomials.
Solution:
(-35m²) + (-78m²) = (-35 + (-78))m² = -113m²
Problem 4: 125nst + (-194nst)
This problem presents the task of summing monomials with multiple variables. We are asked to add 125nst and -194nst. Both terms contain the same variables (n, s, and t) and each variable has an exponent of 1, making them like terms. To add these terms, we add their coefficients: 125 and -194. This involves adding a positive number and a negative number. The result will have the sign of the number with the larger absolute value. The absolute value of -194 is 194, which is greater than 125. Therefore, the result will be negative. Subtracting 125 from 194 gives us 69. Thus, 125 + (-194) equals -69. Consequently, 125nst + (-194nst) simplifies to -69nst. This example illustrates how the principle of adding like terms extends to monomials with multiple variables. As long as the variables and their exponents are the same, the terms can be combined by adding their coefficients. This skill is crucial for simplifying more complex algebraic expressions that involve multiple variables. The solution -69nst demonstrates the correct application of these principles, showing how to handle both positive and negative coefficients in the context of multiple variables.
Solution:
125nst + (-194nst) = (125 + (-194))nst = -69nst
Problem 5: 24lr + 48lr + (-35bp)
In this problem, we encounter the task of summing monomials with a mix of like and unlike terms. The expression is 24lr + 48lr + (-35bp). We observe that 24lr and 48lr are like terms because they both contain the variables l and r raised to the power of 1. The term -35bp, however, is not a like term because it contains the variables b and p, which are different from l and r. To simplify the expression, we first combine the like terms 24lr and 48lr. Adding their coefficients, 24 and 48, gives us 72. Thus, 24lr + 48lr equals 72lr. The term -35bp cannot be combined with 72lr because they are not like terms. Therefore, the simplified expression is 72lr + (-35bp) or 72lr - 35bp. This example highlights the importance of carefully identifying like terms before attempting to add or subtract. It demonstrates that only terms with the same variables raised to the same powers can be combined. The final expression, 72lr - 35bp, consists of two unlike terms, illustrating that simplification sometimes results in an expression with multiple terms that cannot be further combined.
Solution:
24lr + 48lr + (-35bp) = (24 + 48)lr + (-35bp) = 72lr - 35bp
Problem 6: 45x²y + (-27x²y) + 65x²y
This problem involves summing monomials with the same variables and exponents, providing a clear illustration of combining like terms. The expression is 45x²y + (-27x²y) + 65x²y. All three terms contain the variables x raised to the power of 2 and y raised to the power of 1. This means they are like terms and can be combined by adding their coefficients. The coefficients are 45, -27, and 65. To find the sum, we first add 45 and -27, which gives us 18. Then, we add 18 to 65, resulting in 83. Therefore, the sum of the coefficients is 83. Consequently, 45x²y + (-27x²y) + 65x²y simplifies to 83x²y. This example reinforces the concept of adding like terms and demonstrates how to handle multiple terms with both positive and negative coefficients. The key is to ensure that all terms have the same variables and exponents before combining them. The final result, 83x²y, is a single term, showing the simplified form of the original expression. This problem is a fundamental exercise in algebraic simplification, crucial for handling more complex expressions.
Solution:
45x²y + (-27x²y) + 65x²y = (45 - 27 + 65)x²y = 83x²y
Problem 7: (-5df) + 67d²f + (-84df) + (-98d²f)
In this problem, we delve into summing monomials with both like and unlike terms, requiring careful identification and combination. The expression is (-5df) + 67d²f + (-84df) + (-98d²f). We need to identify the like terms before attempting to add them. We have two types of terms: terms with df and terms with d²f. The terms -5df and -84df are like terms because they both contain the variables d and f, each raised to the power of 1. The terms 67d²f and -98d²f are also like terms because they both contain the variables d raised to the power of 2 and f raised to the power of 1. First, let's combine the df terms: -5df + (-84df). Adding the coefficients -5 and -84 gives us -89. Thus, -5df + (-84df) equals -89df. Next, let's combine the d²f terms: 67d²f + (-98d²f). Adding the coefficients 67 and -98 involves subtracting 67 from 98, which gives us 31. Since -98 has a larger absolute value, the result is negative. Thus, 67d²f + (-98d²f) equals -31d²f. Now, we combine the simplified like terms: -89df and -31d²f. Since these terms are not alike (one has d and f, while the other has d² and f), they cannot be combined further. Therefore, the simplified expression is -89df - 31d²f. This example emphasizes the importance of correctly identifying like terms and combining them separately. It also illustrates that the final simplified expression may contain multiple terms if there are unlike terms that cannot be further reduced.
Solution:
(-5df) + 67d²f + (-84df) + (-98d²f) = (-5 - 84)df + (67 - 98)d²f = -89df - 31d²f
Problem 8: 62c³d + (-45cd³) + (-87c³d)
In this final problem, we tackle summing monomials with higher exponents and varying variable arrangements, further solidifying our understanding of like terms. The expression is 62c³d + (-45cd³) + (-87c³d). We need to identify like terms before combining them. The terms 62c³d and -87c³d are like terms because they both contain the variable c raised to the power of 3 and the variable d raised to the power of 1. The term -45cd³ is not a like term because it contains the variable c raised to the power of 1 and the variable d raised to the power of 3, which is different from the exponents in the other terms. To simplify the expression, we first combine the like terms 62c³d and -87c³d. Adding the coefficients 62 and -87 involves subtracting 62 from 87, which gives us 25. Since -87 has a larger absolute value, the result is negative. Thus, 62c³d + (-87c³d) equals -25c³d. The term -45cd³ cannot be combined with -25c³d because they are not like terms. Therefore, the simplified expression is -25c³d + (-45cd³) or -25c³d - 45cd³. This example underscores the critical role of exponents in determining like terms. Even if the variables are the same, terms with different exponents cannot be combined. The final expression, -25c³d - 45cd³, consists of two unlike terms, demonstrating that careful attention to detail is necessary when simplifying algebraic expressions.
Solution:
62c³d + (-45cd³) + (-87c³d) = (62 - 87)c³d + (-45cd³) = -25c³d - 45cd³
By working through these problems, you have gained a solid understanding of how to sum simple monomials. Remember to always identify like terms before adding their coefficients and pay close attention to the variables and their exponents. This skill is fundamental for success in algebra and will be invaluable as you tackle more complex algebraic problems.
