Finding The Value Of C In The Equation 5^5 / 5^2 = A^b = C

In this article, we will delve into the realm of exponential equations, specifically focusing on deciphering the value of 'c' in the equation: $\frac{5^5}{5^2}=a^b=c$. This problem, seemingly straightforward, unveils the fundamental principles of exponents and their manipulation. We will break down the equation step-by-step, providing a clear and concise explanation to ensure a thorough understanding. By the end of this exploration, you will not only be able to solve this specific problem but also gain a solid foundation for tackling similar exponential challenges. Let's embark on this mathematical journey together!

Dissecting the Equation: A Step-by-Step Approach

To determine the value of 'c', we need to meticulously dissect the given equation: $\frac{5^5}{5^2}=a^b=c$. The equation presents us with a series of interconnected expressions. The core of the problem lies in understanding the properties of exponents, particularly the division rule which states that when dividing exponents with the same base, you subtract the powers. We will first focus on simplifying the left-hand side of the equation, $\frac{5^5}{5^2}$, using this rule. This simplification will lead us to a numerical value, which will then be equated to $a^b$ and ultimately reveal the value of 'c'. This methodical approach ensures clarity and accuracy in our solution. The beauty of mathematics lies in its logical progression, and by understanding each step, we can confidently arrive at the correct answer. Our journey begins with the simplification of the exponential expression, laying the groundwork for the subsequent steps in our quest to find the value of 'c'.

Applying the Quotient Rule of Exponents

Our initial focus is on simplifying the expression $\frac{5^5}{5^2}$. Here, the quotient rule of exponents comes into play. This rule dictates that when dividing exponential expressions with the same base, we subtract the exponents. In mathematical terms, this is expressed as: $\frac{x^m}{x^n} = x^{m-n}$. Applying this rule to our expression, we get: $\frac{5^5}{5^2} = 5^{5-2} = 5^3$. This simplification is a crucial step as it transforms the division problem into a more manageable exponential form. Understanding and applying these fundamental rules of exponents is paramount in solving various mathematical problems. The quotient rule is not just a formula to memorize; it's a reflection of the underlying mathematical principles governing exponents. By subtracting the exponents, we are essentially canceling out the common factors in the numerator and denominator. The result, $5^3$, represents the simplified form of the original expression, paving the way for the next stage of our calculation. This step-by-step approach ensures that we grasp the logic behind each operation, fostering a deeper understanding of the mathematical concepts involved.

Evaluating 5 cubed ($5^3$)

Having simplified the expression to $5^3$, the next step is to evaluate this exponential term. The expression $5^3$ signifies 5 raised to the power of 3, which means multiplying 5 by itself three times: $5^3 = 5 * 5 * 5$. Performing this multiplication, we get: $5 * 5 = 25$, and then $25 * 5 = 125$. Therefore, $5^3$ equals 125. This calculation is a straightforward application of the definition of exponents. Understanding the concept of exponents as repeated multiplication is fundamental in mathematics. The exponent indicates the number of times the base is multiplied by itself. In this case, the base is 5, and the exponent is 3, leading to the result 125. This numerical value is a significant milestone in our problem-solving journey, as it directly relates to the value of 'c'. By meticulously evaluating $5^3$, we have bridged the gap between the initial exponential expression and the final solution. This step showcases the importance of accurate calculation and a clear understanding of exponential notation. The value 125 now holds the key to unlocking the final answer to our problem.

Equating to 'c' and Finding the Solution

Now that we have determined that $\frac{5^5}{5^2} = 125$, we can connect this result back to the original equation: $\frac{5^5}{5^2}=a^b=c$. Since we found that $\frac{5^5}{5^2}$ simplifies to 125, and this is equal to 'c', we can directly conclude that c = 125. This step is the culmination of our previous efforts, where each step was carefully executed to arrive at this final value. The equation $a^b = c$ indicates that 125 can also be expressed in exponential form, which in this case is $5^3$. However, the question specifically asks for the value of 'c', and we have definitively determined it to be 125. This direct equivalence is a powerful demonstration of the transitive property of equality, which states that if a = b and b = c, then a = c. In our context, this means that since $\frac{5^5}{5^2}$ equals 125 and $\frac{5^5}{5^2}$ also equals 'c', then 'c' must equal 125. Therefore, the solution to the problem is c = 125. This final step underscores the importance of connecting the intermediate results to the original problem statement to ensure that the answer directly addresses the question being asked. With this conclusive solution, we have successfully navigated the exponential equation and unveiled the value of 'c'.

Conclusion: The Value of 'c' Unveiled

In conclusion, by systematically applying the quotient rule of exponents and meticulously evaluating the resulting expression, we have successfully determined the value of 'c' in the equation $\frac{5^5}{5^2}=a^b=c$. Our journey began with simplifying the exponential expression using the rule $\frac{x^m}{x^n} = x^{m-n}$, which led us to $5^3$. Subsequently, we evaluated $5^3$ as 125. Finally, by equating this result to 'c', we confidently concluded that c = 125. This problem serves as an excellent illustration of the fundamental principles of exponents and their application in solving mathematical equations. The step-by-step approach employed in this solution highlights the importance of understanding each individual operation and its contribution to the overall result. The ability to manipulate exponents and simplify expressions is a crucial skill in mathematics, and this problem provides a solid foundation for further exploration of more complex mathematical concepts. The value of 'c', 125, represents not just a numerical answer, but also a testament to the power of logical reasoning and precise calculation in the realm of mathematics. We have not only solved the problem but also reinforced our understanding of exponential operations. The correct answer is C. 125.

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