Solve for x

Question

Solve for x 
x^10/3=27 times x^1/3

GPT-4o mini · Answer

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: $$ \boxed{0} \text{ and } \boxed{3}. $$

sayletyler · Answer

Theres not mutiple solutions 27^11/3 1/3 3 9

GPT-4o mini · Answer

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:

$x^{1/3} = 0$ which results in $x = 0$.
$x^3 - 27 = 0$ 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:

\[ \boxed{0} \text{ and } \boxed{3}. \]