The provided expression is:

[ \sqrt[3]{a} + 2 \cdot \sqrt[6]{a^2} ]

Solution:

1. Rewrite using fractional exponents:
   • $\sqrt[3]{a} = a^{\frac{1}{3}}$
   • $\sqrt[6]{a^2} = a^{\frac{2}{6}} = a^{\frac{1}{3}}$

Thus, the expression becomes: $a^{\frac{1}{3}} + 2 \cdot a^{\frac{1}{3}}$

2. Combine like terms: Since both terms involve $a^{\frac{1}{3}}$, they can be added: $a^{\frac{1}{3}} + 2 \cdot a^{\frac{1}{3}} = 3 \cdot a^{\frac{1}{3}}$

Final Answer:

$3 \cdot \sqrt[3]{a}$

Would you like a detailed breakdown of any step, or help with related problems?

Related Questions:

1. How do you simplify expressions involving higher roots and powers?
2. What are the rules for combining terms with fractional exponents?
3. Can you provide an example involving mixed radicals like $\sqrt[4]{x^3} + 3\sqrt[4]{x^3}$?
4. How do you convert between radicals and exponents in general?
5. What are the applications of fractional exponents in real-world problems?

Tip:

Always simplify radicals into fractional exponents when combining like terms, as it makes operations like addition or multiplication clearer!