Asked by sayletyler
Solve for x
x^10/3=27 times x^1/3
x^10/3=27 times x^1/3
Answers
Answered by
GPT-4o mini
To solve the equation
\[
x^{10/3} = 27 \cdot x^{1/3},
\]
we start by rewriting \(27\) as \(3^3\):
\[
x^{10/3} = 3^3 \cdot x^{1/3}.
\]
Now, we can rearrange the equation:
\[
x^{10/3} - 3^3 \cdot x^{1/3} = 0.
\]
Next, we can factor out \(x^{1/3}\):
\[
x^{1/3} \left(x^{10/3 - 1/3} - 27\right) = 0.
\]
This simplifies to:
\[
x^{1/3} \left(x^3 - 27\right) = 0.
\]
Now, we have two factors that can be set to zero:
1. \(x^{1/3} = 0\)
2. \(x^3 - 27 = 0\)
From the first factor, \(x^{1/3} = 0\) gives:
\[
x = 0.
\]
From the second factor, \(x^3 - 27 = 0\) gives:
\[
x^3 = 27 \implies x = 27^{1/3} = 3.
\]
Thus, the solutions for \(x\) are:
\[
\boxed{0} \text{ and } \boxed{3}.
\]
\[
x^{10/3} = 27 \cdot x^{1/3},
\]
we start by rewriting \(27\) as \(3^3\):
\[
x^{10/3} = 3^3 \cdot x^{1/3}.
\]
Now, we can rearrange the equation:
\[
x^{10/3} - 3^3 \cdot x^{1/3} = 0.
\]
Next, we can factor out \(x^{1/3}\):
\[
x^{1/3} \left(x^{10/3 - 1/3} - 27\right) = 0.
\]
This simplifies to:
\[
x^{1/3} \left(x^3 - 27\right) = 0.
\]
Now, we have two factors that can be set to zero:
1. \(x^{1/3} = 0\)
2. \(x^3 - 27 = 0\)
From the first factor, \(x^{1/3} = 0\) gives:
\[
x = 0.
\]
From the second factor, \(x^3 - 27 = 0\) gives:
\[
x^3 = 27 \implies x = 27^{1/3} = 3.
\]
Thus, the solutions for \(x\) are:
\[
\boxed{0} \text{ and } \boxed{3}.
\]
Answered by
sayletyler
Theres not mutiple solutions
27^11/3
1/3
3
9
27^11/3
1/3
3
9
Answered by
GPT-4o mini
Let's go through the equation step-by-step and clarify the solutions.
Given the equation:
\[
x^{10/3} = 27 \cdot x^{1/3}
\]
### Step 1: Rewrite the equation
First, note that \(27 = 3^3\). We can rewrite the equation as:
\[
x^{10/3} = 3^3 \cdot x^{1/3}
\]
### Step 2: Move all terms to one side
Rearranging gives:
\[
x^{10/3} - 3^3 \cdot x^{1/3} = 0
\]
### Step 3: Factor out common terms
Factor out \(x^{1/3}\):
\[
x^{1/3}(x^{10/3 - 1/3} - 27) = 0
\]
This simplifies to:
\[
x^{1/3}(x^3 - 27) = 0
\]
### Step 4: Set each factor to zero
This gives us two cases:
1. \(x^{1/3} = 0\) which results in \(x = 0\).
2. \(x^3 - 27 = 0\) which leads to \(x^3 = 27\). Solving this gives:
\[
x = 27^{1/3} = 3.
\]
### Final Solutions
Thus, the solutions are \(x = 0\) and \(x = 3\).
### Clarification of Misunderstanding
If you're looking for specific values such as \(27^{11/3}\), \(1/3\), \(3\), or \(9\), it seems there may be some confusion.
From solving the equation:
- \(x = 3\) is valid.
- The other value \(x = 0\) is also a solution.
If you're looking for specific values to check against possible solutions or confirming calculations, please clarify further.
The valid solutions for the given equation are:
\[
\boxed{0} \text{ and } \boxed{3}.
\]
Given the equation:
\[
x^{10/3} = 27 \cdot x^{1/3}
\]
### Step 1: Rewrite the equation
First, note that \(27 = 3^3\). We can rewrite the equation as:
\[
x^{10/3} = 3^3 \cdot x^{1/3}
\]
### Step 2: Move all terms to one side
Rearranging gives:
\[
x^{10/3} - 3^3 \cdot x^{1/3} = 0
\]
### Step 3: Factor out common terms
Factor out \(x^{1/3}\):
\[
x^{1/3}(x^{10/3 - 1/3} - 27) = 0
\]
This simplifies to:
\[
x^{1/3}(x^3 - 27) = 0
\]
### Step 4: Set each factor to zero
This gives us two cases:
1. \(x^{1/3} = 0\) which results in \(x = 0\).
2. \(x^3 - 27 = 0\) which leads to \(x^3 = 27\). Solving this gives:
\[
x = 27^{1/3} = 3.
\]
### Final Solutions
Thus, the solutions are \(x = 0\) and \(x = 3\).
### Clarification of Misunderstanding
If you're looking for specific values such as \(27^{11/3}\), \(1/3\), \(3\), or \(9\), it seems there may be some confusion.
From solving the equation:
- \(x = 3\) is valid.
- The other value \(x = 0\) is also a solution.
If you're looking for specific values to check against possible solutions or confirming calculations, please clarify further.
The valid solutions for the given equation are:
\[
\boxed{0} \text{ and } \boxed{3}.
\]