Relationship Between A, B, And C When $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$

Introduction

When faced with mathematical problems involving algebraic expressions, especially those with fractional exponents, it's crucial to understand the underlying relationships and identities that can simplify the problem. In this article, we delve into a specific condition: if $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$, and explore the implications and equivalent forms of this equation. This problem is a classic example of how algebraic manipulation and the application of known identities can lead to elegant solutions. Our main focus will be on determining the correct relationship between $a$, $b$, and $c$ given the initial condition. The options presented involve various forms, including sums, cubes, and products of $a$, $b$, and $c$, making it an excellent exercise in algebraic problem-solving. The main keyword here is the relationship between a, b, and c when the sum of their cube roots is zero. Understanding this relationship is key to solving the problem efficiently.

Problem Statement

Given that $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$, we need to determine which of the following options is correct:

a) $a+b+c=0$ b) $(a+b+c)^3=27 a b c$ c) $a+b+c=3(a b c)^{1 / 3}$ d) $a^3+b^3+c^3=0$

This problem tests our ability to manipulate algebraic expressions, particularly those involving cube roots. We'll need to use algebraic identities and logical reasoning to arrive at the correct answer. The core challenge lies in transforming the given equation into a form that reveals the relationship between $a$, $b$, and $c$ as expressed in the options. Algebraic manipulation will be the key technique we employ. We will explore different algebraic pathways to transform the original equation into a form that matches one of the provided options. This process will involve strategic use of identities and careful simplification. The challenge is to identify the correct path and avoid unnecessary complications.

Solution

To solve this problem, let's start by manipulating the given equation. Let $x = a^{1 / 3}$, $y = b^{1 / 3}$, and $z = c^{1 / 3}$. Then the given condition becomes:

$x + y + z = 0$

We want to find a relationship between $a$, $b$, and $c$, which are the cubes of $x$, $y$, and $z$, respectively. A well-known algebraic identity that connects the sum of three terms to their cubes is:

$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$

Since $x + y + z = 0$, the right side of the equation becomes 0:

$x^3 + y^3 + z^3 - 3xyz = 0$

Thus,

$x^3 + y^3 + z^3 = 3xyz$

Now, substitute back $x = a^{1 / 3}$, $y = b^{1 / 3}$, and $z = c^{1 / 3}$:

$(a^{1 / 3})^3 + (b^{1 / 3})^3 + (c^{1 / 3})^3 = 3(a^{1 / 3})(b^{1 / 3})(c^{1 / 3})$

This simplifies to:

$a + b + c = 3(abc)^{1 / 3}$

Therefore, option (c) is correct. Let's analyze why the other options are incorrect or do not necessarily follow from the given condition. Option analysis is crucial for confirming our solution.

• Option (a): $a + b + c = 0$ is not necessarily true. While it might hold in some cases, the derived equation $a + b + c = 3(abc)^{1 / 3}$ shows that $a + b + c$ is equal to three times the cube root of $abc$, not necessarily zero.
• Option (b): $(a + b + c)^3 = 27abc$. If we cube both sides of the equation $a + b + c = 3(abc)^{1 / 3}$, we get $(a + b + c)^3 = (3(abc)^{1 / 3})^3 = 27abc$. So, this option is also correct. Therefore, we have identified another correct option through further manipulation of the derived equation.
• Option (d): $a^3 + b^3 + c^3 = 0$ is not necessarily true. The equation we derived, $a + b + c = 3(abc)^{1 / 3}$, does not directly imply that the sum of the cubes of $a$, $b$, and $c$ is zero.

Thus, after careful analysis and algebraic manipulation, we've determined that options (b) and (c) are the correct answers. The initial equation, $a^{1 / 3} + b^{1 / 3} + c^{1 / 3} = 0$, leads to the derived relationships that confirm these options.

Detailed Explanation of Correct Options

Option (b): $(a+b+c)^3=27abc$

As we derived in the solution, starting from $a^{1/3} + b^{1/3} + c^{1/3} = 0$, we can manipulate this equation using the substitutions $x = a^{1/3}$, $y = b^{1/3}$, and $z = c^{1/3}$. This transforms the original condition into $x + y + z = 0$. We then utilized the algebraic identity $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$. Since $x + y + z = 0$, the identity simplifies to $x^3 + y^3 + z^3 = 3xyz$. Substituting back the original variables, we get $a + b + c = 3(abc)^{1/3}$.

To arrive at option (b), we cube both sides of the equation $a + b + c = 3(abc)^{1/3}$. This yields $(a + b + c)^3 = (3(abc)^{1/3})^3$. Simplifying the right-hand side, we have $(3(abc)^{1/3})^3 = 3^3 imes ((abc)^{1/3})^3 = 27abc$. Therefore, $(a + b + c)^3 = 27abc$, which confirms that option (b) is indeed a correct deduction from the given condition. This cubing operation is a crucial step in linking the intermediate result to the final form presented in option (b).

This option highlights a crucial algebraic relationship that arises when the sum of cube roots equals zero. It demonstrates how seemingly simple initial conditions can lead to significant and sometimes surprising results. The relationship $(a + b + c)^3 = 27abc$ is a strong indicator of the interdependence between $a$, $b$, and $c$ under the given constraint.

Option (c): $a+b+c=3(a b c)^{1 / 3}$

This option is the direct result of our algebraic manipulation. As explained above, starting with $a^{1/3} + b^{1/3} + c^{1/3} = 0$, we made the substitutions $x = a^{1/3}$, $y = b^{1/3}$, and $z = c^{1/3}$, leading to $x + y + z = 0$. The algebraic identity $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$ then simplifies to $x^3 + y^3 + z^3 = 3xyz$ because $x + y + z = 0$. Substituting back $a^{1/3}$, $b^{1/3}$, and $c^{1/3}$, we obtain $a + b + c = 3(abc)^{1/3}$.

This equation explicitly shows the relationship between the sum of $a$, $b$, and $c$ and their product. It demonstrates that $a + b + c$ is three times the cube root of the product $abc$. This is a direct and elegant result that stems from the initial condition. The directness of this derivation underscores the power of algebraic identities in simplifying complex expressions.

This result is particularly significant as it connects the sum of the variables to their product in a non-trivial way. It's not immediately obvious from the initial condition that such a direct relationship exists, which highlights the importance of algebraic manipulation in revealing hidden connections. The equation $a + b + c = 3(abc)^{1/3}$ provides valuable insight into the nature of the relationship between $a$, $b$, and $c$ under the given constraint.

Why Other Options Are Incorrect

Option (a): $a+b+c=0$

This option is not necessarily true. While it might hold for specific values of $a$, $b$, and $c$, it is not a general consequence of the given condition $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$. As we derived, the correct relationship is $a + b + c = 3(abc)^{1/3}$. This equation clearly shows that $a + b + c$ is not always zero; it is zero only when $3(abc)^{1/3}$ is zero, which implies $abc = 0$. Therefore, $a + b + c = 0$ only if one or more of $a$, $b$, or $c$ is zero. Incorrect generalization is the key reason why this option is not universally true.

For example, consider $a = 1$, $b = 1$, and $c = -8$. Then $a^{1/3} = 1$, $b^{1/3} = 1$, and $c^{1/3} = -2$, so $a^{1/3} + b^{1/3} + c^{1/3} = 1 + 1 - 2 = 0$. However, $a + b + c = 1 + 1 - 8 = -6$, which is not zero. This counterexample effectively demonstrates that option (a) is not a valid general conclusion.

Option (d): $a^3+b^3+c^3=0$

This option is also not necessarily true. There is no direct algebraic manipulation that leads from the given condition $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$ to the conclusion that $a^3 + b^3 + c^3 = 0$. The equation we derived, $a + b + c = 3(abc)^{1/3}$, does not provide any immediate insight into the sum of the cubes of $a$, $b$, and $c$. The lack of a direct connection between the derived equation and this option makes it incorrect.

To see why this is the case, consider the example used earlier: $a = 1$, $b = 1$, and $c = -8$. We know that $a^{1/3} + b^{1/3} + c^{1/3} = 0$. However, $a^3 + b^3 + c^3 = 1^3 + 1^3 + (-8)^3 = 1 + 1 - 512 = -510$, which is clearly not zero. This serves as another counterexample confirming that option (d) is not a valid general conclusion.

Conclusion

In summary, given the condition $a^{1 / 3}+b^{1 / 3}+c^{1 / 3}=0$, we have shown that options (b) $(a+b+c)^3=27 a b c$ and (c) $a+b+c=3(a b c)^{1 / 3}$ are correct. We achieved this by using algebraic manipulation and applying a key algebraic identity. The incorrectness of options (a) and (d) was demonstrated through logical reasoning and counterexamples. The key takeaway is the power of algebraic manipulation and identity application in solving mathematical problems. Understanding how to transform and simplify equations is crucial in mathematics.

This problem exemplifies how a seemingly simple condition can lead to significant algebraic relationships. It emphasizes the importance of recognizing and applying relevant algebraic identities to uncover these relationships. The solution process involves careful substitution, simplification, and logical deduction, which are fundamental skills in mathematical problem-solving. The ability to identify and utilize the appropriate algebraic tools is a hallmark of mathematical proficiency. Ultimately, the combination of algebraic technique and logical reasoning is what allows us to navigate complex mathematical challenges and arrive at accurate conclusions.