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Let $x = 2^{1/3}$. Then $4^{1/3} = (2^2)^{1/3} = (2^{1/3})^2 = x^2$.
\n" ); document.write( "So, $a = x^2 + x + 1$.
\n" ); document.write( "We want to find the value of $\frac{3}{a} + \frac{3}{a^2} + \frac{1}{a^3}$.
\n" ); document.write( "Let's find a common denominator:
\n" ); document.write( "$$ \frac{3a^2 + 3a + 1}{a^3} $$
\n" ); document.write( "We know that $a = x^2 + x + 1$. Multiply both sides by $x-1$:
\n" ); document.write( "$$ a(x-1) = (x^2+x+1)(x-1) = x^3 - 1 $$
\n" ); document.write( "Since $x = 2^{1/3}$, we have $x^3 = 2$.
\n" ); document.write( "So, $a(x-1) = 2 - 1 = 1$.
\n" ); document.write( "Therefore, $x-1 = \frac{1}{a}$, which means $x = 1 + \frac{1}{a}$.
\n" ); document.write( "Now, $x^3 = 2$, so
\n" ); document.write( "$$ \left(1 + \frac{1}{a}\right)^3 = 2 $$
\n" ); document.write( "$$ 1 + \frac{3}{a} + \frac{3}{a^2} + \frac{1}{a^3} = 2 $$
\n" ); document.write( "$$ \frac{3}{a} + \frac{3}{a^2} + \frac{1}{a^3} = 2 - 1 $$
\n" ); document.write( "$$ \frac{3}{a} + \frac{3}{a^2} + \frac{1}{a^3} = 1 $$\r
\n" ); document.write( "\n" ); document.write( "Thus, the value of $\frac{3}{a} + \frac{3}{a^2} + \frac{1}{a^3}$ is 1.\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{1}$
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