https://file.aiquickdraw.com/mathbot/file/1726526662364xuyk734i.jpg

The expression in the image is:

\[
-2p^3 \sqrt[3]{648p^5q^2}
\]

Let me simplify this step by step:

### Step 1: Simplify the cube root.
We are dealing with \(\sqrt[3]{648p^5q^2}\).

1. Factor \(648\):
   \[
   648 = 2^3 \times 3^4
   \]

2. Apply the cube root to \(648p^5q^2\):
   \[
   \sqrt[3]{648p^5q^2} = \sqrt[3]{2^3 \times 3^4 \times p^5 \times q^2}
   \]
   Now, simplify the cube root:
   \[
   \sqrt[3]{2^3} = 2, \quad \sqrt[3]{3^3} = 3, \quad \sqrt[3]{3} = \sqrt[3]{3}
   \]
   For the variables:
   \[
   \sqrt[3]{p^5} = p^{5/3}, \quad \sqrt[3]{q^2} = q^{2/3}
   \]

So, the expression becomes:
\[
2 \times 3 \times \sqrt[3]{3} \times p^{5/3} \times q^{2/3} = 6 \sqrt[3]{3} p^{5/3} q^{2/3}
\]

### Step 2: Substitute this back into the original expression.
Now, substitute this into the original expression:
\[
-2p^3 \times 6 \sqrt[3]{3} p^{5/3} q^{2/3}
\]

### Step 3: Combine the terms.
Simplify the powers of \(p\):
\[
p^3 \times p^{5/3} = p^{9/3 + 5/3} = p^{14/3}
\]
So the expression becomes:
\[
-12 p^{14/3} q^{2/3} \sqrt[3]{3}
\]

This is the simplified form of the given expression.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How would you simplify higher roots with multiple variables?
2. What is the general rule for combining powers of the same base?
3. Can this expression be simplified further for specific values of \( p \) and \( q \)?
4. How does this expression change if the exponent on \( p \) or \( q \) increases?
5. How would this expression behave if the cube root was replaced by a square root?

**Tip:** When simplifying expressions involving roots and powers, always handle constants and variables separately to avoid errors.

Learn how to simplify the expression -2p^3 * ∛(648p^5q^2) using exponent rules and radical simplification. This step-by-step guide covers cube roots and exponents, suitable for students in Grades 10-12.

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