Equivalent Expressions: Simplify $10x^9$ And $(60x^{-6})^{-1}$

In this article, we'll dive into simplifying algebraic expressions. Our main focus will be on identifying two expressions that are equivalent to the product of $10x^9$ and $(60x^{-6})^{-1}$, with the crucial assumption that $x$ is not equal to 0. This exploration will involve understanding the rules of exponents and how to manipulate them effectively. Our given options include $\frac{x^{15}}{6}$ and $6x^{15}$, and we'll systematically break down the problem to determine which, if either, are correct. By working through this problem, we'll not only find the solution but also reinforce key algebraic principles. These principles are foundational in mathematics, and mastering them is essential for anyone delving into more advanced topics. Simplifying expressions is a core skill that allows for problem-solving in various contexts, making it a valuable tool in your mathematical toolkit. Let's embark on this journey of simplification and discovery together!

Understanding the Problem

To solve this problem effectively, we must first break down the given expression. We have the product of $10x^9$ and $(60x^{-6})^{-1}$. The goal is to simplify this product and then compare the simplified form with the provided options: $\frac{x^{15}}{6}$ and $6x^{15}$. This task involves a solid understanding of exponent rules, especially how to handle negative exponents and the power of a product. The negative exponent in $(60x^{-6})^{-1}$ indicates that we'll need to take the reciprocal of the base and change the sign of the exponent. Furthermore, we need to remember that when we have a power raised to another power, we multiply the exponents. This rule will be critical when simplifying $(60x^{-6})^{-1}$. Accurately applying these rules is paramount to arriving at the correct answer. Mistakes in exponent manipulation are common, so careful attention to detail is essential. This step-by-step approach will not only help us find the correct expression but also reinforce our understanding of algebraic manipulation. This foundational skill is crucial for more complex mathematical problems, making it worthwhile to master. The initial step of understanding the problem clearly sets the stage for a successful simplification process.

Step-by-Step Simplification

Now, let's go through the simplification process step-by-step. Our original expression is the product of $10x^9$ and $(60x^{-6})^{-1}$. The first step is to simplify $(60x^{-6})^{-1}$. When we have a term raised to a negative power, we take the reciprocal of the term and change the exponent's sign. So, $(60x^{-6})^{-1}$ becomes $\frac{1}{60x^{-6}}$. Next, we can move the term with the negative exponent in the denominator, $x^{-6}$, to the numerator by changing the sign of the exponent. This gives us $\frac{x^6}{60}$. Now, we multiply this simplified term by $10x^9$. So, we have $10x^9 \cdot \frac{x^6}{60}$. To multiply these terms, we multiply the coefficients (10 and $\frac{1}{60}$) and add the exponents of the variable $x$. Multiplying the coefficients gives us $\frac{10}{60}$, which simplifies to $\frac{1}{6}$. Adding the exponents of $x$ gives us $x^{9+6}$, which is $x^{15}$. Therefore, the simplified expression is $\frac{1}{6}x^{15}$ or $\frac{x^{15}}{6}$. This systematic simplification highlights the importance of following the rules of exponents meticulously to avoid errors. Each step builds upon the previous one, leading to the final simplified form. Mastering these steps is fundamental to success in algebra.

Comparing with the Options

After simplifying the original expression $10x^9 \cdot (60x^{-6})^{-1}$, we arrived at the simplified form $\frac{x^{15}}{6}$. Now, we need to compare this result with the options provided: $\frac{x^{15}}{6}$ and $6x^{15}$. A direct comparison immediately reveals that our simplified expression, $\frac{x^{15}}{6}$, matches the first option exactly. The second option, $6x^{15}$, is significantly different from our result. The coefficient is different; in our simplified expression, the coefficient of $x^{15}$ is $\frac{1}{6}$, while in the second option, it is 6. This difference arises from the correct application of the exponent rules and simplification steps we performed earlier. Recognizing these differences is crucial in determining the correct answer. It underscores the importance of not just simplifying but also accurately interpreting the simplified form in the context of the given options. This comparative analysis solidifies our understanding of the simplification process and confirms our solution. By carefully matching our result with the options, we ensure that our answer is both mathematically correct and logically sound.

Conclusion: Choosing the Correct Expressions

In conclusion, after simplifying the expression $10x^9 \cdot (60x^{-6})^{-1}$, we found that it is equivalent to $\frac{x^{15}}{6}$. When we compared this simplified form with the given options, $\frac{x^{15}}{6}$ and $6x^{15}$, it became clear that only the first option, $\frac{x^{15}}{6}$, matches our result. Therefore, $\frac{x^{15}}{6}$ is one of the correct expressions. Since the question asks for two expressions, and we only have two options provided, we can confidently say that $6x^{15}$ is not a correct expression. This exercise has not only helped us identify the correct expression but also reinforced the importance of following the rules of exponents and algebraic simplification. The step-by-step approach we took ensured accuracy and clarity in our solution. Mastering these algebraic techniques is crucial for success in more advanced mathematical studies. The ability to simplify expressions and compare them effectively is a fundamental skill that will benefit you in various mathematical contexts. By understanding and applying these principles, you can tackle similar problems with confidence and precision.

For further learning on algebraic expressions and exponent rules, you can visit Khan Academy for comprehensive resources.