Okay, let's see. The problem is asking for the value of (16 times the cube root of a squared) raised to the power of one-third, given that a is 8. Hmm, let me break this down step by step. 

First, I need to substitute a with 8 in the expression. So, a² would be 8 squared. Let me calculate that. 8 squared is 64. Then, the cube root of 64. The cube root of 64 is the number which, when multiplied by itself three times, gives 64. Wait, 4 times 4 is 16, 16 times 4 is 64. So, the cube root of 64 is 4. Got that part. 

So, now the expression inside the parentheses is 16 multiplied by 4. Let me compute that. 16 times 4 is 64. So, the expression simplifies to 64 raised to the power of one-third. 

Now, the cube root of 64 is again 4 because 4 cubed is 64. Wait, but here it's 64 to the power of one-third. So, that's the same as taking the cube root of 64, which is 4. Therefore, the entire expression simplifies to 4. 

Wait, let me verify that again to make sure I didn't skip any steps or make a mistake. Starting with a = 8. Then a² is 64. Cube root of 64 is 4. Multiply that by 16: 16 * 4 = 64. Then, 64^(1/3) is 4. Yep, that seems right. 

Alternatively, maybe I can approach this using exponent rules. Let's see. The original expression is (16 * ∛(a²))^(1/3). Let's rewrite the cube root as a power of 1/3. So, ∛(a²) is the same as a^(2/3). Then the entire expression becomes (16 * a^(2/3))^(1/3). 

Using exponent rules, when you raise a product to a power, each factor is raised to that power. So, 16^(1/3) * (a^(2/3))^(1/3). Simplifying the exponents: 16^(1/3) times a^(2/3 * 1/3) = a^(2/9). 

But since a is 8, let's substitute that in. 16^(1/3) is the cube root of 16. Wait, 16 is 2^4, so cube root of 2^4 is 2^(4/3). Then, a^(2/9) is 8^(2/9). 8 is 2^3, so 8^(2/9) is (2^3)^(2/9) = 2^(6/9) = 2^(2/3). 

So, putting it all together: 2^(4/3) * 2^(2/3) = 2^(4/3 + 2/3) = 2^(6/3) = 2^2 = 4. 

Okay, that matches my previous result. So, both methods give me 4. That gives me more confidence that the answer is correct. 

Alternatively, maybe I can check using logarithms. Let's take the natural logarithm of the expression. ln((16 * ∛(8²))^(1/3)) = (1/3) * ln(16 * ∛(64)). Then, ln(16) + ln(∛64) = ln(16) + ln(4) because ∛64 is 4. Then, ln(16) is ln(2^4) = 4 ln 2, and ln(4) is ln(2^2) = 2 ln 2. So, total inside the log is 4 ln 2 + 2 ln 2 = 6 ln 2. Multiply by 1/3: (6 ln 2)/3 = 2 ln 2. Then, exponentiate to get e^(2 ln 2) = e^(ln 2^2) = 2^2 = 4. 

Same answer again. So, regardless of the method—direct computation, exponent rules, logarithms—the result is 4. 

I think that's thorough. I considered multiple approaches to verify the result. All paths lead to 4, so I can be confident that's the correct answer.
Thus, the value of $\left(16\sqrt[3]{a^2}\right)^{\frac 13}$ when $a = 8$ is $\boxed{4}$.
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