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

1 answer

To solve the equation

\[ \frac{x^{10}}{3} = 27 x^{\frac{1}{3}}, \]

we can first eliminate the fraction by multiplying both sides by 3:

\[ x^{10} = 81 x^{\frac{1}{3}}. \]

Next, we can rearrange this equation to isolate one side:

\[ x^{10} - 81 x^{\frac{1}{3}} = 0. \]

Now, we can factor out \( x^{\frac{1}{3}} \):

\[ x^{\frac{1}{3}}(x^{\frac{29}{3}} - 81) = 0. \]

Setting each factor to zero gives us two cases to solve:

  1. \( x^{\frac{1}{3}} = 0 \)
  2. \( x^{\frac{29}{3}} - 81 = 0 \)

From the first case, \( x^{\frac{1}{3}} = 0 \) implies:

\[ x = 0. \]

From the second case:

\[ x^{\frac{29}{3}} = 81. \]

We can rewrite 81 as \( 3^4 \):

\[ x^{\frac{29}{3}} = 3^4. \]

Taking both sides to the power of \( \frac{3}{29} \):

\[ x = (3^4)^{\frac{3}{29}} = 3^{\frac{12}{29}}. \]

Thus, the complete solution for \( x \) is:

\[ x = 0 \quad \text{or} \quad x = 3^{\frac{12}{29}}. \]

To summarize, the solutions are:

\[ \boxed{0} \quad \text{and} \quad \boxed{3^{\frac{12}{29}}}. \]