Simplifying The Expression 14z^4y^5 / 7z^5y^7 A Step-by-Step Guide

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to manipulate and rewrite complex expressions into more manageable forms, making them easier to understand and work with. This article delves into the process of simplifying algebraic expressions, using the example of $\frac{14 z^4 y^5}{7 z^5 y^7}$ to illustrate the key concepts and techniques involved. We will break down the expression step by step, explaining the rules of exponents and how they apply to simplifying such expressions. By the end of this guide, you will have a solid understanding of how to simplify algebraic expressions and be able to tackle similar problems with confidence.

Understanding the Basics of Algebraic Expressions

Before diving into the specifics of simplifying the given expression, let's establish a clear understanding of what algebraic expressions are and the basic principles that govern them. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown quantities, while constants are fixed numerical values. The rules of algebra dictate how these components interact, allowing us to manipulate and simplify expressions while maintaining their mathematical equivalence. One of the core concepts in simplifying algebraic expressions is the understanding of exponents. Exponents indicate the number of times a base is multiplied by itself. For instance, in the term $x^3$, $x$ is the base and $3$ is the exponent, meaning $x$ is multiplied by itself three times ($x \* x \* x$). When simplifying expressions with exponents, we often encounter rules such as the quotient rule, which states that when dividing terms with the same base, we subtract the exponents. This rule is crucial for simplifying expressions like the one we are addressing in this article. Furthermore, it's important to remember the order of operations (PEMDAS/BODMAS), which dictates the sequence in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By adhering to these fundamental principles, we can systematically simplify algebraic expressions and arrive at the correct solutions.

Step-by-Step Simplification of $\frac{14 z^4 y^5}{7 z^5 y^7}$

Let's embark on a detailed, step-by-step simplification of the expression $\frac{14 z^4 y^5}{7 z^5 y^7}$. This process will not only provide the solution but also illuminate the underlying principles of simplifying algebraic expressions with exponents. First, we can address the coefficients, which are the numerical values in front of the variables. In this case, we have 14 in the numerator and 7 in the denominator. Dividing 14 by 7, we get 2. So, the expression can be partially simplified to $\frac{2 z^4 y^5}{z^5 y^7}$. Next, we turn our attention to the variables with exponents. Here, we apply the quotient rule of exponents, which states that when dividing terms with the same base, we subtract the exponents. For the variable $z$, we have $z^4$ in the numerator and $z^5$ in the denominator. Subtracting the exponents (4 - 5) gives us -1. Therefore, the $z$ term simplifies to $z^{-1}$. Similarly, for the variable $y$, we have $y^5$ in the numerator and $y^7$ in the denominator. Subtracting the exponents (5 - 7) gives us -2. Thus, the $y$ term simplifies to $y^{-2}$. Now our expression looks like $2 z^{-1} y^{-2}$. However, it is conventional to express exponents as positive values. To do this, we use the rule that $x^{-n} = \frac{1}{x^n}$. Applying this rule to our expression, we move $z^{-1}$ and $y^{-2}$ to the denominator, changing the signs of the exponents. This gives us the final simplified form: $\frac{2}{z^1 y^2}$ or simply $\frac{2}{z y^2}$. This step-by-step approach illustrates how to systematically simplify algebraic expressions by addressing coefficients and variables separately, applying the rules of exponents, and ensuring that the final expression is in its simplest form.

Applying the Quotient Rule of Exponents

A cornerstone of simplifying algebraic expressions, especially those involving division, is the quotient rule of exponents. This rule is essential for efficiently handling expressions like $\frac{14 z^4 y^5}{7 z^5 y^7}$. The quotient rule states that when dividing terms with the same base, you subtract the exponent in the denominator from the exponent in the numerator. Mathematically, this is expressed as $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents. To illustrate this rule in action, let's revisit our expression. We have $\frac{14 z^4 y^5}{7 z^5 y^7}$. First, we separate the coefficients and the variables: $\frac{14}{7} \* \frac{z^4}{z^5} \* \frac{y^5}{y^7}$. Dividing the coefficients, we get 2. Now, applying the quotient rule to the $z$ terms, we have $\frac{z^4}{z^5} = z^{4-5} = z^{-1}$. Similarly, for the $y$ terms, we have $\frac{y^5}{y^7} = y^{5-7} = y^{-2}$. So, our expression becomes $2 z^{-1} y^{-2}$. As discussed earlier, negative exponents indicate reciprocals, so we rewrite $z^{-1}$ as $\frac{1}{z}$ and $y^{-2}$ as $\frac{1}{y^2}$. Combining these, we get $2 \* \frac{1}{z} \* \frac{1}{y^2} = \frac{2}{z y^2}$. This example clearly demonstrates the power and efficiency of the quotient rule in simplifying expressions. By understanding and applying this rule correctly, you can significantly reduce the complexity of algebraic expressions and arrive at the simplest form. The quotient rule is not just a mathematical trick; it's a fundamental property of exponents that simplifies calculations and provides a clearer understanding of the relationships between variables and their powers.

Dealing with Negative Exponents

In the process of simplifying algebraic expressions, encountering negative exponents is quite common, particularly after applying the quotient rule. Understanding how to handle these negative exponents is crucial for expressing the final answer in its most simplified and conventional form. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. In mathematical terms, $x^{-n}$ is equivalent to $\frac{1}{x^n}$. This principle is the key to transforming terms with negative exponents into terms with positive exponents, which are generally preferred in simplified expressions. Let's consider the intermediate result we obtained while simplifying $\frac{14 z^4 y^5}{7 z^5 y^7}$: $2 z^{-1} y^{-2}$. Here, both $z$ and $y$ have negative exponents. To eliminate these negative exponents, we apply the rule mentioned above. The term $z^{-1}$ becomes $\frac{1}{z^1}$ or simply $\frac{1}{z}$, and the term $y^{-2}$ becomes $\frac{1}{y^2}$. Now, substituting these back into the expression, we get $2 \* \frac{1}{z} \* \frac{1}{y^2}$. Combining these terms, we arrive at the simplified form $\frac{2}{z y^2}$. This example highlights the straightforward process of dealing with negative exponents. By recognizing that a negative exponent signifies a reciprocal, you can easily rewrite the term with a positive exponent in the denominator. This transformation is essential for presenting algebraic expressions in their simplest and most widely accepted form. Moreover, understanding negative exponents is not just about following a rule; it deepens your understanding of the relationship between exponents and the multiplicative inverse, a fundamental concept in algebra.

Identifying the Correct Answer

Having meticulously simplified the expression $\frac{14 z^4 y^5}{7 z^5 y^7}$, it's time to identify the correct answer from the given options. Our step-by-step simplification led us to the result $\frac{2}{z y^2}$. Now, let's examine the provided options and see which one matches our simplified expression.

• A. $\frac{2 y^4}{z^4}$: This option is incorrect. It has both $y$ and $z$ in the wrong positions and with incorrect exponents.
• B. $\frac{7 y^3}{z^2}$: This option is also incorrect. The coefficient is wrong, and the exponents of the variables do not match our simplified expression.
• C. $7 x^4 y^4$: This option is incorrect. It involves different variables ($x$ instead of $z$) and is not a simplified form of the original expression.
• D. $2 x^2 y^3$: This option is incorrect as well. It also uses different variables ($x$ instead of $z$) and does not match our simplified expression.

Upon closer inspection, we realize that none of the provided options exactly match our simplified answer of $\frac{2}{z y^2}$. This discrepancy highlights the importance of carefully working through each step of the simplification process and double-checking the final result. It's possible that there was an error in the provided options, or perhaps the question was designed to test the ability to simplify the expression rather than just match an answer. In such cases, it's crucial to be confident in your own work and recognize when the given options are incorrect. Our thorough simplification process confirms that $\frac{2}{z y^2}$ is indeed the correct simplified form of the given expression.

Common Mistakes to Avoid

Simplifying algebraic expressions can sometimes be tricky, and it's easy to fall prey to common mistakes if you're not careful. Being aware of these pitfalls can help you avoid errors and ensure accurate results. One frequent mistake is misapplying the quotient rule of exponents. Remember, this rule applies only when dividing terms with the same base. For instance, $\frac{x^5}{x^2}$ simplifies to $x^{5-2} = x^3$, but it cannot be applied to terms with different bases, such as $\frac{x^5}{y^2}$. Another common error is mishandling negative exponents. It's crucial to remember that a negative exponent indicates a reciprocal, not a negative number. So, $x^{-3}$ is equal to $\frac{1}{x^3}$, not $-x^3$. Forgetting this distinction can lead to incorrect simplifications. Another pitfall is incorrectly combining terms. Only like terms (terms with the same variable and exponent) can be added or subtracted. For example, $3x^2$ and $5x^2$ can be combined to give $8x^2$, but $3x^2$ and $5x^3$ cannot be combined. Furthermore, mistakes often arise from neglecting the order of operations (PEMDAS/BODMAS). Failing to perform operations in the correct sequence can lead to drastically different results. For example, in the expression $2 + 3 \* 4$, multiplication should be performed before addition, yielding $2 + 12 = 14$, not $5 \* 4 = 20$. Lastly, careless arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can derail the entire simplification process. It's always wise to double-check your calculations to minimize these errors. By being mindful of these common mistakes and practicing careful, systematic simplification, you can significantly improve your accuracy and confidence in algebraic manipulations.

Conclusion: Mastering Algebraic Simplification

In conclusion, mastering the art of simplifying algebraic expressions is a fundamental skill in mathematics. It not only enables you to solve complex problems but also enhances your understanding of mathematical relationships and structures. Throughout this comprehensive guide, we have explored the step-by-step process of simplifying expressions, focusing on the example of $\frac{14 z^4 y^5}{7 z^5 y^7}$. We delved into the quotient rule of exponents, a crucial tool for simplifying expressions involving division, and learned how to handle negative exponents effectively. We also emphasized the importance of avoiding common mistakes, such as misapplying the quotient rule, mishandling negative exponents, incorrectly combining terms, neglecting the order of operations, and committing arithmetic errors. By understanding these potential pitfalls, you can approach algebraic simplification with greater confidence and accuracy. While none of the provided options in our example matched the correct simplified form, this situation served as a valuable reminder to trust your own work and critically evaluate the given choices. Always double-check your steps and ensure that your final answer is in its simplest form. With consistent practice and a solid grasp of the underlying principles, you can confidently tackle a wide range of algebraic expressions and excel in your mathematical endeavors. Remember, simplification is not just about finding the right answer; it's about developing a deeper understanding of the language of mathematics and its elegant simplicity.