We are tasked with selecting the expressions that are equal to $\frac{a^4 \sqrt{bc^3}}{a^2b^{-2}c}$. Let's first simplify this expression step by step:

Step 1: Simplify the numerator

The numerator is $a^4 \sqrt{bc^3}$.

• $\sqrt{bc^3}$ can be rewritten as $\sqrt{b} \cdot \sqrt{c^3} = \sqrt{b} \cdot c^{3/2}$.
• Therefore, the numerator becomes $a^4 \sqrt{b} \cdot c^{3/2}$.

Step 2: Simplify the denominator

The denominator is $a^2b^{-2}c$, which can be left as it is for now.

Step 3: Combine the two

Now, divide the numerator by the denominator: $\frac{a^4 \sqrt{b} \cdot c^{3/2}}{a^2b^{-2}c} = a^{4-2} \cdot \frac{\sqrt{b}}{b^{-2}} \cdot \frac{c^{3/2}}{c}$ Simplifying:

• $a^{4-2} = a^2$
• $\frac{\sqrt{b}}{b^{-2}} = b^{1/2 + 2} = b^{5/2}$
• $\frac{c^{3/2}}{c} = c^{3/2 - 1} = c^{1/2}$

Thus, the simplified expression is: $a^2 b^{5/2} c^{1/2}$

Now let's evaluate the options to see which ones are equal to this expression:

Option A: $\frac{a^2b^{5/2}}{c^2}$

• This has the correct form for $a^2b^{5/2}$, but the power of $c$ is incorrect (it has $c^{-2}$ instead of $c^{1/2}$). So, A is not correct.

Option B: $\frac{c^2b^{5/2}}{a^{-2}}$

• This expression can be rewritten as $a^2 b^{5/2} c^2$, but the power of $c$ is incorrect (it has $c^2$ instead of $c^{1/2}$). So, B is not correct.

Option C: $\sqrt{a^4b^5c^4}$

• This simplifies to $a^2 b^{5/2} c^2$, but the power of $c$ is incorrect (it has $c^2$ instead of $c^{1/2}$). So, C is not correct.

Option D: $(abc)^2 \sqrt{b}$

• This simplifies to $a^2 b^2 c^2 \sqrt{b} = a^2 b^{5/2} c^2$, but the power of $c$ is incorrect (it has $c^2$ instead of $c^{1/2}$). So, D is not correct.

Conclusion:

None of the given options match the expression $a^2 b^{5/2} c^{1/2}$.

Do you have any questions or need further clarification?

Here are 5 related questions:

1. How do you simplify expressions with fractional exponents?
2. How do you divide powers with the same base?
3. What are the steps to simplify a radical expression?
4. How do you handle negative exponents in an equation?
5. How do you simplify products and quotients of roots?

Tip: Always check the powers of all variables when comparing algebraic expressions to ensure accuracy.