Dividing Exponential Expressions Simplify (12t^6s^9)/(6ts^7)

July 10, 2025 by Scholario Team 61 views

Dividing Exponential Expressions with Positive Exponents

Dividing exponential expressions is a fundamental concept in algebra, particularly when dealing with variables raised to various powers. In this comprehensive guide, we will delve into the intricacies of dividing expressions involving exponents, focusing on simplifying expressions and ensuring all answers are expressed with positive exponents. This is a crucial skill in algebra, often encountered in various mathematical contexts, from basic equation solving to more advanced calculus problems. Understanding the rules and techniques for handling exponents is essential for mastering algebraic manipulations and achieving accuracy in mathematical calculations.

The foundation of dividing exponential expressions lies in the quotient rule of exponents. This rule states that when dividing two expressions with the same base, you subtract the exponents. 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. This rule is a direct consequence of the definition of exponents as repeated multiplication. When you divide, you are essentially canceling out common factors in the numerator and the denominator. The quotient rule simplifies the division process, allowing us to efficiently handle expressions with exponents. To effectively use this rule, it’s critical to accurately identify the base and the exponents in the expression. Once identified, the subtraction of the exponents needs to be performed with care, paying attention to the signs and ensuring that the base remains the same throughout the operation. This foundational understanding of the quotient rule is key to simplifying a wide range of exponential expressions and is a building block for more complex algebraic manipulations.

Let's tackle the problem at hand: ${ \frac{12 t^6 s^9}{6 t s^7} }$ . This expression involves dividing terms with exponents. The first step in simplifying this expression is to address the coefficients, which are the numerical parts of the terms. In this case, we have 12 in the numerator and 6 in the denominator. Dividing 12 by 6 gives us 2. This is a straightforward arithmetic operation that simplifies the numerical component of the expression. Next, we turn our attention to the variables, which are ${ t }$ and ${ s }$ , each raised to certain powers. For the variable ${ t }$ , we have ${ t^6 }$ in the numerator and ${ t }$ in the denominator, which can be written as ${ t^1 }$ . Applying the quotient rule, we subtract the exponents: 6 minus 1, resulting in ${ t^5 }$ . Similarly, for the variable ${ s }$ , we have ${ s^9 }$ in the numerator and ${ s^7 }$ in the denominator. Subtracting the exponents, 9 minus 7, gives us ${ s^2 }$ . By addressing the coefficients and applying the quotient rule to each variable, we break down the complex expression into simpler components, making it easier to manage and simplify. This step-by-step approach is crucial for accurately simplifying any exponential expression.

Combining these results, we get ${ 2t^5s^2 }$ . This is the simplified form of the original expression, where all exponents are positive. Positive exponents are crucial in mathematical notation because they represent repeated multiplication in a clear and straightforward manner. An expression with positive exponents is generally considered to be in its simplest form. Ensuring that all exponents are positive often involves manipulating negative exponents, which we will discuss later. In this case, both ${ t }$ and ${ s }$ have positive exponents, indicating that the simplification is complete according to the given requirement. The final simplified expression, ${ 2t^5s^2 }$ , is not only mathematically correct but also adheres to the standard convention of expressing exponential terms with positive exponents. This convention is important for clear communication in mathematics, ensuring that expressions are easily understood and interpreted by others.

To provide a detailed understanding, let’s break down the solution into a step-by-step process:

Divide the coefficients: Divide the numerical coefficients in the numerator and the denominator. This is a basic arithmetic operation that simplifies the overall expression. In our example, we divide 12 by 6 to get 2. This step is crucial as it separates the numerical component from the variable components, making the simplification process more manageable. By addressing the coefficients first, we reduce the complexity of the expression, allowing us to focus on the variables and their exponents. This approach is particularly helpful in more complex expressions where multiple terms and variables are involved. The result of this step provides a solid foundation for the subsequent steps in simplifying the expression.
Apply the quotient rule for exponents: For each variable, subtract the exponent in the denominator from the exponent in the numerator. This is the core application of the quotient rule of exponents. For the variable ${ t }$ , we subtract the exponent 1 from 6, resulting in ${ t^5 }$ . For the variable ${ s }$ , we subtract the exponent 7 from 9, resulting in ${ s^2 }$ . This step is where the essence of exponential division comes into play. It requires a clear understanding of the rule and careful attention to the exponents involved. The subtraction must be performed accurately, ensuring that the correct result is obtained for each variable. This process effectively simplifies the variable components of the expression, leading us closer to the final simplified form. The consistent application of the quotient rule is key to successfully dividing exponential expressions.
Combine the results: Multiply the simplified coefficient with the simplified variable terms. This step brings together the results from the previous steps, combining the numerical coefficient with the simplified variable expressions. In our case, we multiply 2 with ${ t^5 }$ and ${ s^2 }$ to get ${ 2t^5s^2 }$ . This step is crucial for presenting the final simplified expression in a coherent form. It ensures that all the components are properly integrated, resulting in a single term that represents the original expression in its simplest form. The combination of the coefficient and the variable terms should be done carefully, ensuring that the variables are written in a standard order, typically alphabetical. This final combination step completes the simplification process, providing the answer in a clear and concise manner.
Ensure positive exponents: Check if all exponents are positive. If any exponent is negative, rewrite the term by moving it to the opposite side of the fraction (numerator to denominator or vice versa). This step is essential for adhering to the requirement of expressing answers with positive exponents. Negative exponents indicate the reciprocal of the base raised to the positive value of the exponent. For instance, ${ a^{-n} }$ is equivalent to ${ \frac{1}{a^n} }$ . In our example, all exponents are already positive, so no further action is needed. However, in many problems, this step is critical for ensuring the final answer is in the correct form. The process of converting negative exponents to positive exponents involves understanding the relationship between the base and its exponent and applying the rules of reciprocals. This final check ensures that the expression is not only simplified but also presented in the standard mathematical notation with positive exponents.

Sometimes, after applying the quotient rule, you might end up with negative exponents. Remember, a term with a negative exponent can be rewritten using the rule ${ a^{-n} = \frac{1}{a^n} }$ . This means you move the term to the opposite side of the fraction (numerator to denominator or vice versa) and change the sign of the exponent. Negative exponents represent the reciprocal of the base raised to the positive exponent. Understanding this relationship is crucial for simplifying expressions and ensuring that all exponents are positive in the final answer. When dealing with negative exponents, it is important to identify them correctly and apply the reciprocal rule accurately. This involves moving the base and its exponent to the opposite side of the fraction, which effectively changes the sign of the exponent. This process is not just about changing the sign; it’s about changing the meaning of the term to its reciprocal form. This step is often necessary to fully simplify an expression and present it in the standard mathematical form with positive exponents. The ability to handle negative exponents is a key skill in algebra and is essential for solving a wide range of problems.

For instance, if we had ${ \frac{t^2}{t^5} }$ , applying the quotient rule would give us ${ t^{2-5} = t^{-3} }$ . To rewrite this with a positive exponent, we would express it as ${ \frac{1}{t^3} }$ . This transformation is a direct application of the negative exponent rule. The ability to move between negative and positive exponents is a fundamental skill in simplifying algebraic expressions. It allows for flexibility in manipulating equations and is essential for solving more complex problems. The process involves a clear understanding of the reciprocal relationship and the proper application of the rule. By ensuring that all exponents are positive, we are adhering to the standard mathematical notation and making the expression easier to understand and interpret. This skill is not only important for simplifying expressions but also for various applications in algebra and beyond.

To solidify your understanding, let’s look at a few more examples:

Simplify ${ \frac{15x^8y^3}{5x^2y^5} }$ . In this problem, we have two variables, ${ x }$ and ${ y }$ , each raised to different powers. The first step is to divide the coefficients, 15 divided by 5, which gives us 3. Next, we apply the quotient rule to each variable. For ${ x }$ , we subtract the exponent 2 from 8, resulting in ${ x^6 }$ . For ${ y }$ , we subtract the exponent 5 from 3, resulting in ${ y^{-2} }$ . Notice that ${ y }$ has a negative exponent. To express this with a positive exponent, we rewrite ${ y^{-2} }$ as ${ \frac{1}{y^2} }$ . Combining these results, we get ${ \frac{3x^6}{y^2} }$ . This example demonstrates the importance of handling negative exponents and rewriting them to ensure the final answer is in the required form.
Simplify ${ \frac{8a^4b^6}{16a^7b^2} }$ . Here, we again have two variables, and the coefficients need to be simplified. Dividing 8 by 16 gives us ${ \frac{1}{2} }$ . For the variable ${ a }$ , we subtract the exponent 7 from 4, resulting in ${ a^{-3} }$ . For the variable ${ b }$ , we subtract the exponent 2 from 6, resulting in ${ b^4 }$ . To express ${ a }$ with a positive exponent, we rewrite ${ a^{-3} }$ as ${ \frac{1}{a^3} }$ . Combining these results, we get ${ \frac{b^4}{2a^3} }$ . This example further illustrates how to deal with coefficients that simplify to fractions and how to handle negative exponents that arise during the simplification process.

Dividing exponential expressions requires a solid grasp of the quotient rule and the ability to handle negative exponents. By following the step-by-step process outlined in this guide, you can confidently simplify a wide range of expressions and ensure your answers are expressed with positive exponents. The key to success is understanding the underlying principles, practicing regularly, and paying close attention to detail. Mastering these skills will not only help you in algebra but also in various other fields of mathematics and science where exponential expressions are commonly encountered. The ability to simplify expressions efficiently and accurately is a valuable asset in any mathematical endeavor. By consistently applying these techniques, you will enhance your problem-solving abilities and gain a deeper understanding of algebraic concepts.