Simplify The Mathematical Expression $-25 B^6 + 26 B^6$.

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Introduction

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. Simplifying expressions not only makes them easier to understand but also facilitates further calculations and problem-solving. This article delves into the simplification of the algebraic expression $-25b^6 + 26b^6$. We will explore the step-by-step process, underlying principles, and the significance of this simplification in broader mathematical contexts. Whether you are a student, educator, or math enthusiast, this guide will provide a comprehensive understanding of how to approach and simplify such expressions effectively.

Understanding the Basics of Algebraic Expressions

Before diving into the specifics of the given expression, it’s crucial to grasp the basic components of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. In the expression $-25b^6 + 26b^6$, we have constants (-25 and 26), a variable (b), and an exponent (6). The term $b^6$ represents the variable 'b' raised to the power of 6, meaning 'b' multiplied by itself six times. Understanding these components is the first step in simplifying any algebraic expression. Recognizing like terms is also essential; like terms are those that have the same variable raised to the same power. In our expression, both terms have $b^6$, making them like terms, which means they can be combined through addition or subtraction.

Key Components of Algebraic Expressions

• Variables: These are symbols (usually letters) that represent unknown values. In our expression, 'b' is the variable.
• Constants: These are fixed numerical values. Here, -25 and 26 are constants.
• Exponents: An exponent indicates the number of times the base is multiplied by itself. In $b^6$, 6 is the exponent.
• Coefficients: The numerical factor of a term that contains a variable. -25 and 26 are the coefficients of the terms $-25b^6$ and $26b^6$, respectively.

Step-by-Step Simplification of $-25b^6 + 26b^6$

To simplify the expression $-25b^6 + 26b^6$, we need to identify and combine like terms. As mentioned earlier, like terms are terms that have the same variable raised to the same power. In this case, both $-25b^6$ and $26b^6$ are like terms because they both contain the variable 'b' raised to the power of 6. The simplification process involves adding or subtracting the coefficients of these like terms while keeping the variable and exponent the same. This is akin to combining similar objects; for instance, you can combine 25 apples and 26 apples because they are the same type of fruit.

Combining Like Terms: The Process

1. Identify Like Terms: In the expression $-25b^6 + 26b^6$, the terms $-25b^6$ and $26b^6$ are like terms.
2. Combine Coefficients: Add or subtract the coefficients of the like terms. In this case, we add -25 and 26.
3. Perform the Operation: $-25 + 26 = 1$. So, the new coefficient is 1.
4. Write the Simplified Term: Combine the new coefficient with the variable and exponent. The simplified term is $1b^6$.
5. Final Simplification: Since 1 as a coefficient is often omitted, the simplified expression is $b^6$.

Detailed Explanation of Each Step

Let’s break down the simplification process into even more detail to ensure clarity and understanding. The ability to simplify algebraic expressions hinges on a solid grasp of the distributive property and the rules of combining like terms. Each step in the simplification process serves a specific purpose, ensuring the expression is reduced to its most basic form without altering its mathematical value.

1. Identifying Like Terms

The first step in simplifying any algebraic expression is to identify the like terms. Like terms are those terms that have the same variable raised to the same power. For example, $3x^2$ and $-5x^2$ are like terms because they both have the variable 'x' raised to the power of 2. On the other hand, $3x^2$ and $3x^3$ are not like terms because the exponents are different. Similarly, $3x^2$ and $3y^2$ are not like terms because the variables are different. In the expression $-25b^6 + 26b^6$, both terms have the same variable 'b' raised to the same power 6, making them like terms.

2. Combining Coefficients

Once like terms are identified, the next step is to combine their coefficients. The coefficient is the numerical factor of a term that contains a variable. For instance, in the term $5x$, 5 is the coefficient. To combine like terms, you simply add or subtract their coefficients. This is based on the distributive property, which allows us to factor out the common variable part. In the expression $-25b^6 + 26b^6$, the coefficients are -25 and 26. We need to perform the operation $-25 + 26$.

3. Performing the Operation

The operation $-25 + 26$ is a simple arithmetic problem. When adding a negative number to a positive number, you are essentially finding the difference between the two numbers and using the sign of the larger number. In this case, the difference between 26 and 25 is 1, and since 26 is the larger number and it is positive, the result is +1. Therefore, $-25 + 26 = 1$. This result becomes the new coefficient of the simplified term.

4. Writing the Simplified Term

After performing the operation on the coefficients, the next step is to write the simplified term. This involves combining the new coefficient with the variable and exponent that were common to the like terms. In our example, the new coefficient is 1, and the common variable and exponent are $b^6$. Thus, the simplified term is $1b^6$. This means 1 multiplied by $b^6$.

5. Final Simplification

The final step in the simplification process is to present the expression in its most concise form. In mathematics, a coefficient of 1 is often omitted because multiplying a term by 1 does not change its value. Therefore, $1b^6$ is usually written simply as $b^6$. This is the most simplified form of the original expression $-25b^6 + 26b^6$. The omission of the coefficient 1 makes the expression cleaner and easier to work with in further calculations.

The Simplified Expression: $b^6$

After following the step-by-step process, the simplified form of the expression $-25b^6 + 26b^6$ is $b^6$. This means that by combining the like terms and performing the necessary arithmetic, we have reduced the original expression to a single term. The process of simplification not only makes the expression more concise but also easier to understand and manipulate in further mathematical operations.

Significance of Simplification

The importance of simplifying algebraic expressions extends beyond mere aesthetics. Simplified expressions are:

• Easier to understand: A simplified expression presents the relationship between variables and constants more clearly.
• Easier to work with: Simplified expressions reduce the likelihood of errors in subsequent calculations.
• Essential for solving equations: Simplification is often a necessary step in solving algebraic equations.

In many mathematical contexts, simplifying expressions is a prerequisite for further analysis and problem-solving. For instance, when solving equations, simplifying both sides of the equation can reveal the solution more readily. In calculus, simplified expressions make differentiation and integration more straightforward. Therefore, mastering the skill of simplifying algebraic expressions is crucial for success in various branches of mathematics.

Common Mistakes to Avoid

When simplifying algebraic expressions, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and ensure accurate simplifications. Here are some common mistakes to watch out for:

1. Incorrectly Combining Unlike Terms: One of the most common errors is combining terms that are not like terms. Remember, terms must have the same variable raised to the same power to be combined. For example, $3x^2$ and $2x$ cannot be combined because they have different powers of 'x'. Similarly, $4y$ and $4z$ cannot be combined because they have different variables.
2. Forgetting the Sign of Coefficients: It’s crucial to pay close attention to the signs (positive or negative) of the coefficients when combining like terms. For instance, in the expression $5x - 3x$, the operation is subtraction, so the result is $2x$. Neglecting the negative sign can lead to errors.
3. Misapplying the Distributive Property: The distributive property is essential for simplifying expressions involving parentheses. A common mistake is not distributing the multiplication across all terms inside the parentheses. For example, $2(x + 3)$ should be simplified to $2x + 6$, not $2x + 3$.
4. Incorrect Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) is vital. Exponents should be dealt with before multiplication or division, and addition and subtraction should be done last. For example, in the expression $3 + 2 imes 4$, multiplication should be done before addition, resulting in $3 + 8 = 11$, not $5 imes 4 = 20$.
5. Errors with Exponents: Mistakes with exponents can easily occur if the rules of exponents are not correctly applied. For example, $(x^2)^3$ is $x^6$, not $x^5$. Similarly, $x^2 imes x^3$ is $x^5$, not $x^6$. Make sure to review and understand the rules of exponents to avoid these errors.

Practice Problems

To reinforce your understanding of simplifying algebraic expressions, let’s work through a few practice problems. These examples will help you apply the concepts discussed and improve your skills in algebraic manipulation.

Practice Problem 1

Simplify the expression: $7a^3 - 4a^3 + 2a^3$

Solution:

1. Identify Like Terms: All three terms ($7a^3$, $-4a^3$, and $2a^3$) are like terms because they have the same variable 'a' raised to the power of 3.
2. Combine Coefficients: Add and subtract the coefficients: $7 - 4 + 2$.
3. Perform the Operation: $7 - 4 = 3$, and $3 + 2 = 5$. The new coefficient is 5.
4. Write the Simplified Term: Combine the new coefficient with the variable and exponent: $5a^3$.

The simplified expression is $5a^3$.

Practice Problem 2

Simplify the expression: $-9x^4 + 5x^4 - x^4$

Solution:

1. Identify Like Terms: All three terms ($-9x^4$, $5x^4$, and $-x^4$) are like terms.
2. Combine Coefficients: Add and subtract the coefficients: $-9 + 5 - 1$.
3. Perform the Operation: $-9 + 5 = -4$, and $-4 - 1 = -5$. The new coefficient is -5.
4. Write the Simplified Term: Combine the new coefficient with the variable and exponent: $-5x^4$.

The simplified expression is $-5x^4$.

Practice Problem 3

Simplify the expression: $12y^2 - 6y^2 + 3y - y$

Solution:

1. Identify Like Terms: We have two sets of like terms: $12y^2$ and $-6y^2$, and $3y$ and $-y$.
2. Combine Coefficients for $y^2$ terms: $12 - 6 = 6$. The simplified term is $6y^2$.
3. Combine Coefficients for $y$ terms: $3 - 1 = 2$. The simplified term is $2y$.
4. Write the Simplified Expression: Combine the simplified terms: $6y^2 + 2y$.

The simplified expression is $6y^2 + 2y$.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics that involves combining like terms to reduce an expression to its simplest form. By systematically identifying like terms, combining their coefficients, and paying attention to the rules of exponents and order of operations, we can effectively simplify complex expressions. In the case of $-25b^6 + 26b^6$, the simplified expression is $b^6$. This process not only makes expressions easier to understand but also facilitates further mathematical manipulations and problem-solving. Mastery of this skill is essential for success in algebra and other advanced mathematical disciplines.

By understanding the step-by-step process and practicing regularly, you can become proficient in simplifying algebraic expressions and enhance your overall mathematical abilities. Remember to always double-check your work and be mindful of common mistakes to ensure accuracy. With consistent effort, simplifying algebraic expressions will become second nature, allowing you to tackle more complex mathematical challenges with confidence.