select all expressions that are equal to a^4sqrootbc^3/a^2b^-2c. A: a^2b^5/2/c^2. B: c^2b^5/2/a^-2. C: sqroot a^4b^5c^4. D: (abc)^2sqrootb



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.

Select all expressions that are equal to a^4sqrootbc^3/a^2b^-2c. A: a^2b^5/2/c^2. B: c^2b^5/2/a^-2. C: sqroot a^4b^5c^4. D: (abc)^2sqrootb.

Learn how to simplify the algebraic expression a^4 √(bc^3) / a^2b^-2c with step-by-step guidance. This problem involves exponent rules, simplifying radicals, and algebraic manipulation. Suitable for students in Grades 9-12.

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