Given the equation \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \), let's explore the implications and properties of this relationship. We'll analyze the expression step by step:
### Step 1: Understanding the Given Equation
The equation \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \) states that the sum of the cube roots of three numbers \( a \), \( b \), and \( c \) is zero.
### Step 2: Derivative with Respect to \( a \)
To explore the behavior of the equation, we take the partial derivative of the expression with respect to \( a \):
[tex]\[ \frac{d}{da} \left( a^{1/3} + b^{1/3} + c^{1/3} \right) \][/tex]
When we compute this derivative, we get:
[tex]\[ \frac{d}{da} \left( a^{1/3} \right) = \frac{1}{3} a^{-2/3} = \frac{1}{3a^{2/3}} \][/tex]
### Step 3: Examining the Derivative
The derivative with respect to \( a \):
[tex]\[ \frac{1}{3a^{2/3}} \][/tex]
This tells us how the expression changes as \( a \) changes. Specifically, it decays as a power of \( a \) when \( a \) is positive.
### Step 4: Solving the Given Equation for \( a \)
Next, we solve the equation \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \) for \( a \). This involves isolating \( a \) and expressing it in terms of \( b \) and \( c \):
[tex]\[ a^{1/3} = - ( b^{1/3} + c^{1/3} ) \][/tex]
Cubing both sides:
[tex]\[ a = \left( - ( b^{1/3} + c^{1/3} ) \right)^3 \][/tex]
Since \( -1 \) cubed is still \( -1 \):
[tex]\[ a = - ( b^{1/3} + c^{1/3} )^3 \][/tex]
### Step 5: Solutions
Therefore, the solutions satisfying \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \) can be written as:
[tex]\[ a = - ( b^{1/3} + c^{1/3} )^3 \][/tex]
with \( b \) and \( c \) remaining as they are.
### Summary
The observations from the conditions given by the equation are:
1. The derivative of \( a^{1/3} + b^{1/3} + c^{1/3} \) with respect to \( a \) is \( \frac{1}{3a^{2/3}} \).
2. Solving the equation for \( a \) yields \( a = - ( b^{1/3} + c^{1/3} )^3 \).
So, the detailed exploration confirms that:
[tex]\[ a^{1/3} + b^{1/3} + c^{1/3} = 0 \][/tex] holds true if and only if [tex]\( a = - ( b^{1/3} + c^{1/3} )^3 \)[/tex].
