Simplifying $(\frac{7}{9} A)(\frac{4}{5} A)$ A Step-by-Step Guide

Simplify the expression $(\frac{7}{9} a)(\frac{4}{5} a)$?

In mathematics, simplifying expressions is a fundamental skill. It involves rewriting an expression in its most compact and manageable form. This is particularly crucial in algebra, where we deal with variables and coefficients. In this article, we will delve into simplifying algebraic expressions, focusing on a specific example: $(\frac{7}{9} a)(\frac{4}{5} a)$. We will break down the steps involved, explain the underlying concepts, and provide additional insights to enhance your understanding. To effectively simplify algebraic expressions, it’s essential to grasp the core principles of arithmetic and algebra. These include the commutative, associative, and distributive properties, as well as the rules for multiplying fractions and variables. These principles provide the foundation for manipulating expressions while maintaining their mathematical integrity. In our specific case, we're dealing with the multiplication of two fractional terms, each containing a variable. This requires us to understand how to multiply fractions and how to handle variables in multiplication. The process may seem straightforward, but a clear understanding of each step is vital for mastering more complex algebraic simplifications. This article aims to provide that clarity, making the process accessible and understandable for learners of all levels. By the end of this guide, you'll be equipped with the knowledge to tackle similar problems and a solid foundation for further algebraic studies. We'll also touch on common pitfalls to avoid, ensuring that you can simplify expressions accurately and efficiently. Our journey begins with a detailed explanation of each step, followed by practical tips and concluding with a summary of the key takeaways.

Step-by-Step Simplification

To simplify the expression $(\frac{7}{9} a)(\frac{4}{5} a)$, we will break it down into manageable steps. The core concept here is the multiplication of fractions and variables. The given expression involves the product of two terms, each comprising a fraction and a variable ‘a’. Let’s dive into the process step by step. First, we need to understand how to multiply fractions. To multiply two fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This is a fundamental rule in arithmetic and is the backbone of our simplification process. In our case, the fractions are $\frac{7}{9}$ and $\frac{4}{5}$. So, we multiply the numerators 7 and 4, and then multiply the denominators 9 and 5. This will give us a new fraction that represents the product of the original fractions. Next, we consider the variables. In the expression, we have ‘a’ multiplied by ‘a’. When multiplying variables with the same base, you add their exponents. In this case, ‘a’ is raised to the power of 1 in both terms (since $a = a^1$). Therefore, when we multiply $a$ by $a$, we are essentially multiplying $a^1$ by $a^1$. Adding the exponents (1 + 1) gives us $a^2$. This rule is a cornerstone of algebraic manipulation and is crucial for simplifying expressions involving variables. Now, let’s combine the results of multiplying the fractions and the variables. We will multiply the simplified fraction we obtained earlier by the simplified variable term ($a^2$). This final step brings together the numerical and algebraic parts of the expression, giving us the simplified form. It's important to note that the order of multiplication doesn't affect the result, thanks to the commutative property of multiplication. This property allows us to rearrange the terms in any order without changing the outcome. In essence, simplifying this expression involves breaking it down into its components, applying the rules of fraction and variable multiplication, and then combining the results. This step-by-step approach makes the process clear and easy to follow, even for those new to algebra. By mastering these fundamental steps, you can confidently tackle more complex algebraic expressions.

Step 1: Multiply the Fractions

The initial step in simplifying the expression $(\frac{7}{9} a)(\frac{4}{5} a)$ is to multiply the numerical fractions. This involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers) of the fractions. Understanding how to multiply fractions is fundamental to simplifying many algebraic expressions, so let's break it down. To multiply fractions, you simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. This rule applies regardless of the complexity of the fractions involved. In our specific case, we have the fractions $\frac{7}{9}$ and $\frac{4}{5}$. The numerators are 7 and 4, and the denominators are 9 and 5. So, we multiply 7 by 4 to get the new numerator, and 9 by 5 to get the new denominator. Multiplying the numerators (7 * 4) gives us 28. This is the numerator of our resulting fraction. Next, we multiply the denominators (9 * 5), which gives us 45. This is the denominator of our resulting fraction. Therefore, when we multiply $\frac{7}{9}$ by $\frac{4}{5}$, we get the fraction $\frac{28}{45}$. This fraction represents the numerical part of our simplified expression. It's crucial to ensure that the fraction is in its simplest form. This means checking if the numerator and denominator have any common factors that can be cancelled out. In this case, 28 and 45 do not share any common factors other than 1, so the fraction $\frac{28}{45}$ is already in its simplest form. This step highlights the importance of basic arithmetic skills in algebra. A solid understanding of fraction multiplication is essential for simplifying algebraic expressions effectively. By correctly multiplying the fractions, we've taken the first significant step towards simplifying the entire expression. This process not only simplifies the numerical part but also sets the stage for handling the variable part of the expression. The next steps will build upon this foundation, further simplifying the expression to its most compact form.

Step 2: Multiply the Variables

The next crucial step in simplifying the expression $(\frac{7}{9} a)(\frac{4}{5} a)$ involves multiplying the variables. In this case, we are multiplying 'a' by 'a'. Understanding how to handle variables in algebraic expressions is fundamental, and this step provides a clear example of the process. When multiplying variables with the same base, such as 'a' in this instance, we use the rule of exponents. This rule states that when multiplying like bases, you add the exponents. In simpler terms, when you multiply $a$ by $a$, you are essentially multiplying $a^1$ by $a^1$ (since any variable without an explicit exponent is understood to have an exponent of 1). To apply the rule, we add the exponents: 1 + 1 = 2. This means that $a$ multiplied by $a$ equals $a^2$. The term $a^2$ is read as "a squared" and represents 'a' multiplied by itself. This is a common concept in algebra and is used extensively in various mathematical contexts. Multiplying variables is a core skill in simplifying algebraic expressions. It allows us to combine like terms and reduce the complexity of an expression. In our specific case, multiplying 'a' by 'a' simplifies the variable component of the expression significantly. This step not only simplifies the variable part but also prepares us to combine the numerical and variable parts of the expression. By correctly applying the rule of exponents, we've taken another important step towards simplifying the entire expression. This process demonstrates the power of algebraic rules in making complex expressions more manageable. It's important to remember this rule as you encounter more complex algebraic problems. The ability to multiply variables correctly is essential for success in algebra and beyond. In the next step, we will combine the results of multiplying the fractions and the variables to arrive at the final simplified expression.

Step 3: Combine the Results

Having simplified both the fractional and variable parts of the expression $(\frac{7}{9} a)(\frac{4}{5} a)$ individually, the final step is to combine these results. This involves bringing together the simplified numerical fraction and the simplified variable term to form the complete simplified expression. In Step 1, we multiplied the fractions $\frac{7}{9}$ and $\frac{4}{5}$ and obtained $\frac{28}{45}$. This fraction represents the numerical coefficient of our final expression. In Step 2, we multiplied the variables 'a' by 'a' and obtained $a^2$. This term represents the variable part of our final expression. Now, to combine these results, we simply multiply the fraction $\frac{28}{45}$ by the variable term $a^2$. This gives us the expression $\frac{28}{45} a^2$. This expression is the simplified form of the original expression $(\frac{7}{9} a)(\frac{4}{5} a)$. It represents the product of the numerical coefficient and the variable term. This step highlights the importance of breaking down complex problems into smaller, manageable parts. By simplifying the fractional and variable parts separately, we were able to tackle the problem systematically and arrive at the solution. Combining the results is a crucial step in simplifying any algebraic expression. It brings together all the simplified components to form the final, most compact form of the expression. This final form is easier to work with and understand. In this case, $\frac{28}{45} a^2$ is a much simpler and clearer representation of the original expression. This step also reinforces the connection between arithmetic and algebra. The rules of fraction multiplication and the rules of exponents work together to simplify the expression. This combination of skills is essential for success in algebra. By mastering the process of combining results, you can confidently simplify a wide range of algebraic expressions. This skill is invaluable in mathematics and its applications in various fields. The ability to simplify expressions efficiently and accurately is a key component of mathematical proficiency.

Final Answer

After following the step-by-step simplification process, the final simplified form of the expression $(\frac{7}{9} a)(\frac{4}{5} a)$ is $\frac{28}{45} a^2$. This answer represents the product of the simplified numerical fraction and the variable term. The journey to this final answer involved several key steps, each building upon the previous one. We began by multiplying the fractions $\frac{7}{9}$ and $\frac{4}{5}$, which resulted in $\frac{28}{45}$. This step required a solid understanding of fraction multiplication, a fundamental concept in arithmetic. Next, we multiplied the variables 'a' by 'a', which resulted in $a^2$. This step showcased the importance of the rules of exponents in simplifying algebraic expressions. Finally, we combined these results to arrive at the simplified expression $\frac{28}{45} a^2$. This final step brought together the numerical and variable components, providing a clear and concise representation of the original expression. The final answer, $\frac{28}{45} a^2$, is not only mathematically correct but also presented in its simplest form. The fraction $\frac{28}{45}$ cannot be further simplified, and the variable term $a^2$ is already in its most compact form. This underscores the importance of simplifying expressions to their fullest extent. The process of arriving at this final answer highlights the interconnectedness of mathematical concepts. Arithmetic skills, such as fraction multiplication, and algebraic rules, such as the rules of exponents, work together to achieve the desired simplification. Understanding this interconnectedness is crucial for mathematical proficiency. By breaking down the problem into smaller, manageable steps, we were able to simplify the expression effectively. This approach is a valuable strategy for tackling complex mathematical problems. It allows for a systematic and organized approach, making the process less daunting and more accessible. The final answer, $\frac{28}{45} a^2$, is a testament to the power of simplification in mathematics. It transforms a seemingly complex expression into a clear and understandable form, making it easier to work with in further calculations or applications.

Additional Tips for Simplifying Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, and there are several tips and tricks that can make the process more efficient and accurate. These tips range from understanding fundamental properties to recognizing common patterns. Mastering these tips will not only help you simplify expressions more effectively but also deepen your understanding of algebra. One of the most important tips is to understand and apply the order of operations (PEMDAS/BODMAS). This acronym stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following the correct order of operations is essential for arriving at the correct simplified form. Another key tip is to recognize and combine like terms. Like terms are 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. You can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables). Understanding the distributive property is also crucial. The distributive property states that $a(b + c) = ab + ac$. This property allows you to multiply a single term by a group of terms inside parentheses. Applying the distributive property correctly is essential for simplifying expressions that involve parentheses. When dealing with fractions, always simplify fractions before combining terms. This can make the numbers smaller and easier to work with. Simplifying fractions involves dividing both the numerator and the denominator by their greatest common factor. Another helpful tip is to pay attention to signs. A common mistake in simplifying expressions is making errors with positive and negative signs. Take your time and double-check your work to ensure that you have handled the signs correctly. Practice is key to mastering the art of simplifying expressions. The more you practice, the more comfortable and confident you will become. Work through a variety of examples, and don't be afraid to make mistakes. Mistakes are learning opportunities. Finally, always check your work. After you have simplified an expression, take a moment to review your steps and ensure that you have not made any errors. You can also substitute values for the variables to check if your simplified expression is equivalent to the original expression. By following these tips and practicing regularly, you can develop strong skills in simplifying algebraic expressions. This skill will serve you well in mathematics and in many other fields.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes, especially when dealing with complex terms or multiple operations. Being aware of these common pitfalls can help you avoid errors and simplify expressions more accurately. One of the most frequent mistakes is incorrectly applying the order of operations. As mentioned earlier, the order of operations (PEMDAS/BODMAS) is crucial for simplifying expressions correctly. Failing to follow this order can lead to incorrect results. For instance, performing addition before multiplication, or neglecting to address parentheses properly can lead to a wrong simplification. Another common error is incorrectly combining like terms. Remember that like terms must have the same variable raised to the same power. A frequent mistake is to combine terms that are not alike, such as adding $3x$ to $4x^2$. This is incorrect because 'x' and $x^2$ are not like terms. Errors with the distributive property are also common. A typical mistake is to only multiply the term outside the parentheses by the first term inside the parentheses, forgetting to distribute it to all terms within the parentheses. For example, in the expression $2(x + 3)$, a mistake would be to write $2x + 3$ instead of the correct $2x + 6$. Sign errors are another significant source of mistakes. Carelessly handling positive and negative signs can lead to incorrect simplifications. For example, subtracting a negative number can be tricky, and it's essential to remember that subtracting a negative is the same as adding a positive. When dealing with fractions, failing to simplify fractions before combining terms can make the problem more complex than necessary. Simplifying fractions first can reduce the size of the numbers and make the calculations easier. Another mistake is incorrectly applying the rules of exponents. Remember that when multiplying like bases, you add the exponents, and when raising a power to a power, you multiply the exponents. Mixing up these rules can lead to errors. Skipping steps in the simplification process can also be a source of mistakes. While it may seem faster to skip steps, it increases the likelihood of making an error. It's better to write out each step clearly and methodically. Finally, not checking your work is a common mistake. Always take the time to review your steps and ensure that you have not made any errors. You can also substitute values for the variables to check if your simplified expression is equivalent to the original expression. By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy in simplifying algebraic expressions. This attention to detail will serve you well in mathematics and in many other areas.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics, and mastering it requires a solid understanding of basic arithmetic and algebraic principles. Through this article, we have demonstrated a step-by-step approach to simplifying the expression $(\frac{7}{9} a)(\frac{4}{5} a)$, highlighting the key concepts involved. We began by multiplying the fractions, then multiplied the variables, and finally combined the results to arrive at the simplified form: $\frac{28}{45} a^2$. This process underscores the importance of breaking down complex problems into smaller, manageable steps. By tackling each component separately, we can simplify expressions more efficiently and accurately. The article also emphasized the significance of understanding the rules of fraction multiplication and the rules of exponents. These rules are the building blocks of algebraic simplification, and a thorough understanding of them is essential for success in mathematics. Furthermore, we discussed additional tips for simplifying expressions, such as following the order of operations, combining like terms, and applying the distributive property correctly. These tips provide valuable strategies for tackling a wide range of algebraic problems. We also highlighted common mistakes to avoid, such as sign errors, incorrect application of the order of operations, and failing to simplify fractions before combining terms. Being aware of these pitfalls can help you prevent errors and improve your accuracy. Simplifying algebraic expressions is not just about finding the correct answer; it's also about developing problem-solving skills and mathematical reasoning. The ability to simplify expressions effectively is a valuable asset in mathematics and its applications in various fields. By practicing regularly and applying the concepts and tips discussed in this article, you can develop strong skills in simplifying algebraic expressions. This skill will serve you well in your mathematical journey and beyond. Remember, mathematics is a journey, and every step you take, every problem you solve, brings you closer to mastery. So, keep practicing, keep learning, and keep simplifying!