Sum Of Cubes: 125x^9 + 64y^12 Explained
Ever wondered how to express a complex polynomial like $125 x^9+64 y^{12}$ as a sum of cubes? It might seem like a puzzle, but with a little algebraic know-how, it's totally solvable! We'll break down the problem, explore the options, and reveal the correct way to represent this expression as a sum of two perfect cubes. Stick around, because understanding this will give you a fantastic boost in your algebraic skills, especially when dealing with factoring and recognizing patterns in expressions. This isn't just about solving a specific problem; it's about mastering a fundamental concept that pops up frequently in higher mathematics, from calculus to abstract algebra. So, let's dive in and demystify the world of sums of cubes!
Understanding the Sum of Cubes Formula
Before we tackle our specific problem, let's refresh our memory on the sum of cubes formula. This is a fundamental algebraic identity that states: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. While we're not asked to factor this expression in this particular question, recognizing the structure of a sum of cubes is key to solving it. The goal is to find two terms, let's call them '' and '', such that '' equals $125 x^9$ and '' equals $64 y^{12}$. Once we find these '' and '' terms, we'll have successfully written the original expression as a sum of cubes, specifically in the form $(a)^3 + (b)^3$. This process involves understanding how exponents work when you're cubing a term that already has an exponent, which follows the rule $(xm)n = x^{m imes n}$. So, if we have a term like $x^9$, and we want to express it as a cube, we need to find a number '' such that $(xm)3 = x^9$. This means $3m = 9$, so $m = 3$. Therefore, $x^9$ can be written as $(x3)3$. This is the crucial step that allows us to break down the powers in our original expression into perfect cubes. Mastering this exponent rule is indispensable for manipulating algebraic expressions and simplifying complex terms. It's the foundation upon which many other algebraic techniques are built, and a solid grasp of it will make tackling more advanced mathematical concepts feel significantly more accessible. Keep this exponent rule in mind as we move forward, as it's the secret sauce to unlocking this problem.
Deconstructing the First Term: $125 x^9$
Let's start by focusing on the first part of our expression: $125 x^9$. Our objective is to express this as '' for some term ''. First, consider the numerical coefficient, 125. We need to find a number that, when multiplied by itself three times, equals 125. Think about common cubes: $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, and $5^3 = 125$. Aha! So, the cube root of 125 is 5. This means that the numerical part of our '' term will be 5. Now, let's look at the variable part, $x^9$. As we discussed earlier with the exponent rule $(xm)n = x^{m imes n}$, we need to find an exponent '' such that $(xm)3 = x^9$. To do this, we set the exponents equal: $3m = 9$. Solving for '', we get $m = 3$. Therefore, $x^9$ can be rewritten as $(x3)3$. Combining the numerical and variable parts, our '' term is $5 x^3$. So, we can confidently say that $125 x^9$ is equivalent to $(5 x3)3$. This step is critical, as it isolates the base term that, when cubed, produces the first part of our original expression. It's like finding the building block that creates the first component of our sum. This meticulous approach ensures accuracy and builds a strong foundation for solving the rest of the problem. Don't hesitate to pause and verify each step; mathematical precision is all about careful decomposition and reconstruction. This careful examination of each component, the coefficient and the variable's exponent, is precisely how we can deconstruct complex terms into their fundamental cubic forms, paving the way for recognizing the overall structure of the sum of cubes. It requires a keen eye for numerical patterns and a solid understanding of exponential properties.
Deconstructing the Second Term: $64 y^{12}$
Now, let's move on to the second part of our expression: $64 y^{12}$. We need to express this as '' for some term ''. Let's start with the coefficient, 64. We are looking for a number that, when cubed, gives us 64. We already found that $4^3 = 64$. So, the numerical part of our '' term is 4. Next, let's tackle the variable part, $y^{12}$. Using the same exponent rule, $(ym)3 = y^{12}$, we need to find '' such that $3m = 12$. Dividing both sides by 3, we get $m = 4$. This means that $y^{12}$ can be rewritten as $(y4)3$. Putting it all together, our '' term is $4 y^4$. Therefore, $64 y^{12}$ is equivalent to $(4 y4)3$. We have now successfully identified both '' and '' in the form $(a)^3 + (b)^3$. This second deconstruction mirrors the process of the first term, reinforcing the application of the exponent rule and the identification of numerical cube roots. Each step is a building block, and by carefully constructing each part, we ensure that the final expression accurately represents the sum of cubes. This methodical approach is essential in mathematics, where small errors in intermediate steps can cascade into incorrect final answers. Being able to confidently break down and represent terms like $64 y^{12}$ as $(4 y4)3$ is a hallmark of algebraic proficiency. It demonstrates a deep understanding of how powers and coefficients interact within mathematical expressions, making complex problems seem more manageable and revealing the elegant structure hidden within. This methodical decomposition is not just about solving this problem; it's about developing a robust problem-solving toolkit applicable to a wide range of mathematical challenges. The ability to see $(4 y4)3$ hidden within $64 y^{12}$ is a testament to careful observation and practice.
Assembling the Sum of Cubes
We've done the hard work of breaking down both terms. We found that $125 x^9$ is equal to $(5 x3)3$ and $64 y^12}$ is equal to $(4 y4)3$. Now, all we need to do is put them back together in the form of a sum of cubes$ written as a sum of cubes. Looking at the options provided:
A. $(25 x3)3+(4 y4)3$ B. $(5 x3)3+(16 y4)3$ C. $(25 x3)3+(8 y4)3$ D. This is where the calculation leads us to the correct choice.
Comparing our result, $(5 x3)3 + (4 y4)3$, with the given options, we can see that none of the options directly match this form as presented. However, let's re-examine the question and options carefully. The question asks for the expression written as a sum of cubes. Our derivation has shown that $125 x^9 = (5x3)3$ and $64 y^{12} = (4y4)3$. Therefore, $125 x^9 + 64 y^{12} = (5x3)3 + (4y4)3$. It seems there might be a misunderstanding in how the options are presented or interpreted in relation to the question. Let's assume the options are meant to represent the terms that are being cubed, rather than the expanded form. If we look at the base terms: for the first term, the base is $5x^3$, and for the second term, the base is $4y^4$. We need to find an option where the bases are $(5x^3)$ and $(4y^4)$, and these bases are then cubed. Let's re-evaluate the options based on this understanding:
A. Bases are $(25x^3)$ and $(4y^4)$. Cubing the first base gives $(25x3)3 = (52x3)^3 = 56x9$, which is not $125x^9$. Incorrect.
B. Bases are $(5x^3)$ and $(16y^4)$. Cubing the first base gives $(5x3)3 = 125x^9$. Cubing the second base gives $(16y4)3 = (42y4)^3 = 46y{12}$, which is not $64y^{12}$. Incorrect.
C. Bases are $(25x^3)$ and $(8y^4)$. Cubing the first base gives $(25x3)3$, which we already know is incorrect. Cubing the second base gives $(8y4)3 = (23y4)^3 = 29y{12}$, which is not $64y^{12}$. Incorrect.
It appears there might be an issue with the provided options, as our derived sum of cubes, $(5 x3)3 + (4 y4)3$, doesn't directly correspond to any of them. However, if we must choose from the given options, and assuming there's a typo or a misinterpretation in the question's options, let's double check our initial calculations. We confirmed $125 = 5^3$ and $64 = 4^3$. We also confirmed $x^9 = (x3)3$ and $y^{12} = (y4)3$. This means the correct sum of cubes must be $(5x3)3 + (4y4)3$. Let's assume there's a typo in the options and one of them is intended to represent this. Often, in multiple-choice questions, one needs to identify the option that most closely matches the correct form, or identify the correct components. In this case, the correct components are $5x^3$ and $4y^4$. None of the options use both these bases correctly. Let's re-read the question and options one more time. The question asks "What is $125 x^9+64 y^{12}$ written as a sum of cubes?". The options are presented as A, B, C, D. It's possible the options are presenting the result of cubing the bases, or perhaps the question or options have a mistake. Given our solid derivation, $(5x3)3 + (4y4)3$ is the correct representation. If we ignore the potential issue with the options and focus on the mathematical process, we've successfully decomposed the expression. Let's consider if any option represents the correct bases that are cubed. Option A has $25x^3$ and $4y^4$. Option B has $5x^3$ and $16y^4$. Option C has $25x^3$ and $8y^4$. Only Option B correctly identifies $5x^3$ as one of the bases. However, the second base is incorrect. Let's consider another possibility: perhaps the question intends for the options to be the bases themselves, and we need to see which pair, when cubed, gives us the original expression. This is how we've been approaching it. There seems to be a discrepancy. Let's assume for a moment that the question means to ask which option represents the sum of cubes correctly. Our derived form is $(5x3)3 + (4y4)3$. If we had to choose, we'd look for an option that has $5x^3$ and $4y^4$ as the terms being cubed. Since that's not an option, let's reconsider the possibility of a typo in the question or options. If, hypothetically, option A was meant to be $(5x3)3+(4y4)3$, that would be correct. Given the provided options, and the definitive mathematical derivation, it appears there is an error in the question's provided choices. However, if we are forced to select the option that contains correct individual components, option A has $4y^4$, and option B has $5x^3$. This situation highlights the importance of carefully checking all given information and being prepared to identify inconsistencies.
Final Thoughts and Conclusion
After meticulously breaking down the expression $125 x^9+64 y^12}$, we determined that it can be perfectly represented as a sum of cubes$ becomes $(y4)3$). While our derived form $(5 x3)3 + (4 y4)3$ is mathematically sound, it does not directly match any of the provided multiple-choice options (A, B, C, D). This suggests a potential error in the question's options. In a real test scenario, if presented with such a discrepancy, it would be wise to double-check your work, as we have done, and if the derivation remains consistent, flag the question or choose the option that most closely resembles the correct form or contains correct individual components, while acknowledging the discrepancy. The process of recognizing and manipulating expressions into sums of cubes is a vital skill in algebra, crucial for factoring and simplifying more complex polynomial expressions. It allows us to see underlying structures that might not be immediately apparent. Mastering these algebraic identities can significantly enhance problem-solving abilities across various mathematical disciplines. For further exploration into algebraic identities and polynomial manipulation, you can consult resources like Brilliant.org or Khan Academy.
