Simplifying Exponential Expressions Finding Equivalents For (2.3)^5 / (0.9)^4)^3

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to manipulate complex equations into more manageable forms, making them easier to understand and solve. This article delves into the process of simplifying the expression ${\left(\frac{(2.3)^5}{(0.9)^4}\right)^3}$, providing a comprehensive explanation of the steps involved and the underlying mathematical principles. Our exploration will not only reveal the equivalent expression but also enhance your understanding of exponent rules and algebraic manipulation.

Understanding the Problem

At its core, this problem challenges us to simplify an expression involving exponents and fractions. The expression ${\left(\frac{(2.3)^5}{(0.9)^4}\right)^3}$ might seem daunting at first glance, but by applying the rules of exponents, we can systematically break it down. The key here is to recognize that the entire fraction, including both the numerator and the denominator, is raised to the power of 3. This means we need to distribute the exponent to each term within the parentheses.

To begin, let’s dissect the expression. We have a fraction where the numerator is ${(2.3)^5}$ and the denominator is ${(0.9)^4}$. This fraction is then raised to the power of 3. Our goal is to find an equivalent expression among the given options: A. ${2.5^3}$, B. ${2.5^{15}}$, C. ${\frac{(2.3)^8}{(0.9)^7}}$, and D. {\frac{(2.3)^{15}}{(0.9)^{12}}\\. Each option presents a different form, and only one will be the correct simplification. By meticulously applying the power of a power rule, which states that \((a^m)^n = a^{m \times n}}, and the power of a quotient rule, which extends this principle to fractions, we can transform the original expression into its simplest equivalent form. This process not only simplifies the expression but also reinforces our grasp of exponent manipulation, a crucial skill in algebra and beyond. Let's embark on this mathematical journey, simplifying step by step to uncover the correct answer and deepen our understanding of these essential concepts.

Step-by-Step Simplification

The first crucial step in simplifying the expression ${\left(\frac{(2.3)^5}{(0.9)^4}\right)^3}$ is to apply the power of a quotient rule. This rule dictates that when a fraction is raised to a power, you must raise both the numerator and the denominator to that power. Mathematically, this is represented as ${(\frac{a}{b})^n = \frac{a^n}{b^n}}$. Applying this rule to our expression, we get:

${ \left(\frac{(2.3)^5}{(0.9)^4}\right)^3 = \frac{((2.3)^5)^3}{((0.9)^4)^3} }$

Now that we've distributed the outer exponent, we need to simplify the numerator and the denominator separately. This is where the power of a power rule comes into play. This rule states that when you raise a power to another power, you multiply the exponents, which is mathematically expressed as ${(a^m)^n = a^{m \times n}}$. Applying this rule to both the numerator and the denominator, we can further simplify our expression.

For the numerator, we have ${((2.3)^5)^3}$. Applying the power of a power rule, we multiply the exponents 5 and 3, resulting in ${2.3^{5 \times 3} = 2.3^{15}}$. Similarly, for the denominator, we have ${((0.9)^4)^3}$. Applying the same rule, we multiply the exponents 4 and 3, resulting in ${0.9^{4 \times 3} = 0.9^{12}}$. Now, substituting these simplified terms back into our fraction, we get:

${ \frac{((2.3)^5)^3}{((0.9)^4)^3} = \frac{2.3^{15}}{0.9^{12}} }$

This simplified expression, {\frac{(2.3)^{15}}{(0.9)^{12}}\}, represents the equivalent form of the original expression. By meticulously applying the power of a quotient and the power of a power rules, we have successfully transformed the complex expression into a more manageable form. This step-by-step simplification not only provides the correct answer but also demonstrates the importance of understanding and applying exponent rules in algebraic manipulations. In the next section, we will compare this simplified expression with the given options to identify the matching equivalent expression, solidifying our understanding of the process.

Identifying the Equivalent Expression

Having simplified the original expression ${\left(\frac{(2.3)^5}{(0.9)^4}\right)^3}$ to {\frac{(2.3)^{15}}{(0.9)^{12}}\}, the next step is to compare this result with the provided options to identify the correct equivalent expression. The options given are:

• A. ${2.5^3}$
• B. ${2.5^{15}}$
• C. ${\frac{(2.3)^8}{(0.9)^7}}$
• D. {\frac{(2.3)^{15}}{(0.9)^{12}}\}Comparing our simplified expression, {\frac{(2.3)^{15}}{(0.9)^{12}}\}, with the options, it becomes clear that option D, {\frac{(2.3)^{15}}{(0.9)^{12}}\}, is the exact match. The other options do not align with our simplified form. Options A and B involve a base of 2.5, which is not present in our simplified expression. Option C has exponents of 8 and 7 in the numerator and denominator, respectively, which do not match the exponents 15 and 12 in our simplified form.

Therefore, by direct comparison, we can confidently conclude that option D is the equivalent expression. This process highlights the importance of accurate simplification and careful comparison when dealing with mathematical expressions. The ability to systematically apply exponent rules and reduce expressions to their simplest forms is a critical skill in mathematics. This exercise not only demonstrates the solution to a specific problem but also reinforces the broader concept of expression equivalence and the methods used to establish it. Identifying the equivalent expression is not just about finding the right answer; it's about understanding the underlying mathematical principles that govern how expressions can be manipulated and transformed while maintaining their value. This understanding is crucial for more advanced mathematical problem-solving.

Why Other Options Are Incorrect

To fully grasp the solution to the problem, it's essential to understand not only why the correct option is right but also why the other options are incorrect. This deeper analysis reinforces the concepts involved and prevents common errors. Let's examine each incorrect option:

• Option A: ${2.5^3}$

  This option is incorrect because it drastically simplifies the original expression without correctly applying the exponent rules. The expression ${2.5^3}$ doesn't account for the exponents in the numerator and denominator of the original fraction, nor does it correctly apply the outer exponent of 3. There's no mathematical justification for reducing the expression to a single term raised to the power of 3, especially with a base of 2.5, which doesn't directly relate to the bases in the original expression. This option likely results from a misunderstanding of how to distribute exponents across a fraction and apply the power of a power rule. Ignoring the fractional structure and the specific bases (2.3 and 0.9) leads to this incorrect simplification.

• Option B: ${2.5^{15}}$

  Similar to option A, option B, ${2.5^{15}}$, fails to correctly apply the exponent rules to the original expression. While the exponent of 15 might suggest some attempt to apply the power of a power rule (multiplying exponents), the base of 2.5 is still incorrect. The original expression involves bases of 2.3 and 0.9, and there's no valid mathematical operation that would transform these into 2.5. This option might arise from focusing solely on the exponents and neglecting the bases, leading to a distorted simplification. The correct application of exponent rules requires considering both the exponents and the bases, ensuring that the operations performed maintain the integrity of the expression.

• Option C: ${\frac{(2.3)^8}{(0.9)^7}}$

  Option C, ${\frac{(2.3)^8}{(0.9)^7}}$, represents a more nuanced error in applying the exponent rules. This option seems to stem from incorrectly adding exponents instead of multiplying them. The power of a power rule states that ${(a^m)^n = a^{m \times n}}$, not ${a^{m+n}}$. In the original expression, the exponent of 3 outside the parentheses should be multiplied by the exponents inside, not added. For instance, ${(2.3^5)^3}$ should become ${2.3^{15}}$ (5 multiplied by 3), not ${2.3^8}$ (5 plus 3). A similar error occurs in the denominator. This option highlights a common mistake in exponent manipulation, emphasizing the importance of correctly distinguishing between the rules for adding exponents (when multiplying like bases) and multiplying exponents (when raising a power to a power).

By understanding why these options are incorrect, we reinforce the correct application of exponent rules and gain a deeper appreciation for the precision required in algebraic manipulations. This detailed analysis serves as a valuable learning tool, helping to avoid similar errors in future problems.

Conclusion

In conclusion, simplifying mathematical expressions requires a thorough understanding and application of fundamental rules and principles. In the case of the expression ${\left(\frac{(2.3)^5}{(0.9)^4}\right)^3}$, we successfully identified the equivalent expression by meticulously applying the power of a quotient rule and the power of a power rule. Through a step-by-step simplification process, we transformed the original expression into {\frac{(2.3)^{15}}{(0.9)^{12}}\}, which corresponds to option D.

This exercise underscores the importance of mastering exponent rules, which are essential tools in algebra and beyond. The ability to correctly manipulate exponents allows us to simplify complex expressions, making them easier to understand and work with. Moreover, by analyzing why the other options were incorrect, we reinforced the nuances of these rules and highlighted common errors to avoid. The incorrect options served as valuable learning opportunities, emphasizing the need for precision and careful application of mathematical principles.

The process of simplifying expressions is not just about arriving at the correct answer; it's about developing a deeper understanding of mathematical structures and relationships. Each step in the simplification process reveals a facet of these relationships, building a more robust foundation for future mathematical endeavors. This article has not only provided a solution to a specific problem but has also aimed to enhance your overall mathematical literacy, empowering you to tackle similar challenges with confidence and accuracy. The journey of simplifying expressions is a journey of mathematical discovery, where each solved problem adds to our understanding and appreciation of the elegance and power of mathematics.