To solve this problem, we are determining which of the given expressions are equivalent to $16^{\frac{5}{2}}$. Let’s break this step by step:

Step 1: Simplify $16^{\frac{5}{2}}$

The expression $16^{\frac{5}{2}}$ can be interpreted as: $16^{\frac{5}{2}} = \left( 16^{\frac{1}{2}} \right)^5$ Here, $16^{\frac{1}{2}}$ represents the square root of 16: $16^{\frac{1}{2}} = \sqrt{16} = 4$ So: $16^{\frac{5}{2}} = 4^5$

Thus, we need to determine which of the options are equivalent to $4^5$.

Step 2: Evaluate Each Option

Option A: $4^5$

• This is directly equivalent to $4^5$.
• Correct.

Option B: $8^5$

• $8 \neq 4$, and raising 8 to the 5th power does not match $16^{\frac{5}{2}}$.
• Incorrect.

Option C: $\sqrt{16^5}$

• The square root can be rewritten as: $\sqrt{16^5} = \left( 16^5 \right)^{\frac{1}{2}} = 16^{\frac{5}{2}}$
• This matches the original expression.
• Correct.

Option D: $\sqrt[5]{16^2}$

• The fifth root of $16^2$ is: $\sqrt[5]{16^2} = \left( 16^2 \right)^{\frac{1}{5}} = 16^{\frac{2}{5}}$
• This does not match $16^{\frac{5}{2}}$.
• Incorrect.

Option E: $(16^2)(16^{\frac{1}{2}})$

• Using the exponent addition rule $a^m \cdot a^n = a^{m+n}$: $(16^2)(16^{\frac{1}{2}}) = 16^{2 + \frac{1}{2}} = 16^{\frac{5}{2}}$
• This matches the original expression.
• Correct.

Option F: $(16^5)(16^{\frac{1}{2}})$

• Again, applying the exponent rule: $(16^5)(16^{\frac{1}{2}}) = 16^{5 + \frac{1}{2}} = 16^{\frac{11}{2}}$
• This does not match $16^{\frac{5}{2}}$.
• Incorrect.

Final Answer:

The correct options are:

• A: $4^5$
• C: $\sqrt{16^5}$
• E: $(16^2)(16^{\frac{1}{2}})$

Do you need further clarifications or additional details?

Here are 5 related questions you might find helpful:

1. How do you simplify expressions with fractional exponents?
2. What are the rules for adding and multiplying exponents?
3. How can roots and powers be expressed as exponents?
4. What is the relationship between $a^{\frac{m}{n}}$ and radicals?
5. How do you simplify expressions involving products of powers?

Tip: Remember, $a^{\frac{m}{n}}$ means the $n$-th root of $a^m$.