<Question>
If $a = 8$, what is the value of $\left(16\sqrt[3]{a^2}\right)^{\frac 13}$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is asking for the value of (16 times the cube root of a squared) to the power of 1/3, given that a is 8. Hmm, let me break this down step by step. First, I need to substitute the value of a into the expression. Since a is 8, I should compute a squared first.

<prompt_1.1> Substitute the value of \( a = 8 \) into the expression \( 16\sqrt[3]{a^2} \) and calculate \( a^2 \). </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, a squared would be 8 squared, which is 64. Then, the cube root of 64. Wait, the cube root of 64 is 4 because 4 times 4 times 4 is 64. Right, so cube root of 64 is 4. Then, 16 times 4. Let me compute that.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
16 multiplied by 4 is 64. So now, the expression simplifies to 64 raised to the power of 1/3. Hmm, 64 to the 1/3.
Since the cube root of 64 is 4, because 4^3 is 64. Wait, but here it's 64^(1/3).

So, 64^(1/3) is the same as the cube root of 64, which is 4.
So, putting it all together: (16 * cube root(a^2))^(1/3) becomes (16 * 4)^(1/3) which is 64^(1/3) equals 4.

Is that right? Let me check again.

<prompt_2.1> Re-evaluate the expression \( (16\sqrt[3]{a^2})^{\frac 13} \) with \( a = 8 \) to verify the solution. </prompt_2.1>

Alternatively, maybe I can approach this using exponent rules. Let's see. The original expression is (16 * a^(2/3))^(1/3). Let's substitute a = 8 first.

<prompt_2.2> Use exponent rules to simplify the expression \( (16\sqrt[3]{a^2})^{\frac 13} \) with \( a = 8 \). </prompt_2.2>

Alternatively, maybe I can combine the exponents first. Let's see.

<prompt_2.3> Combine the exponents first and then substitute \( a = 8 \) to find the solution. </prompt_2.3>

Wait, but let me make sure there isn't a different interpretation of the original expression. Is the cube root applied to just a^2, or is it possible that the expression is 16 multiplied by the cube root of a, all squared? But according to the original problem, it's written as 16\sqrt[3]{a^2}, so the cube root is over a squared. So, yeah, that's correct.

Another way to check: Let's compute the expression inside out. a is 8.

<prompt_2.4> Compute the expression inside out to verify the solution with \( a = 8 \). </prompt_2.4>

Alternatively, if I use exponent rules all the way: (16 * a^(2/3))^(1/3) = 16^(1/3) * (a^(2/3))^(1/3) = 16^(1/3) * a^(2/9).
But 16 is 2^4, so 16^(1/3) is (2^4)^(1/3) = 2^(4/3). Then, a^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3). Multiplying these together: 2^(4/3) * 2^(2/3) = 2^(6/3) = 2^2 = 4.
</Plan_2>

<Execution_2>
<execution_2.1>
Starting with a = 8, compute a^2: 8^2 = 64. Then cube root of 64 is 4. Multiply by 16: 16*4 = 64. Then take the cube root of 64: 64^(1/3) = 4.
</execution_2.1>

<execution_2.2>
So, a is 8, so a^(2/3) is 8^(2/3). 8 to the power of 2/3. Since 8 is 2 cubed, so 8^(1/3) is 2, then 2 squared is 4. So, 8^(2/3) is 4. Then, 16 times 4 is 64. Then, 64 to the 1/3 is 4.
</execution_2.2>

<execution_2.3>
First, substitute a = 8 into the expression: (16 * 8^(2/3))^(1/3). Now, compute 8^(2/3). Since 8 is 2^3, so 8^(2/3) is (2^3)^(2/3) = 2^(3*(2/3)) = 2^2 = 4. So, 16 * 4 is 64. Then, 64^(1/3) is 4. So, same result.
</execution_2.3>

<execution_2.4>
Compute a squared: 8^2 = 64. Then cube root of 64 is 4. Multiply by 16: 16*4=64. Then take the cube root of 64: 4.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3, execution_2.4:
Alternatively, if I did it all in exponents: (16 * a^(2/3))^(1/3) = (16^(1) * a^(2/3))^(1/3) = 16^(1/3) * a^(2/9).
But 16 is 2^4, so 16^(1/3) is 2^(4/3). a is 8, which is 2^3, so a^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3). Then, multiplying 2^(4/3) * 2^(2/3) = 2^(6/3) = 2^2 = 4. Same answer.

I don't see any mistakes in these steps. Each substitution and exponent application seems correct.
So, the value should be 4.
</Plan_3>

<Execution_3>
No prompt to execute.
</Execution_3>

<Final_answer>
Therefore, the value of the given expression is $\boxed{4}$.
</Final_answer>
</think>