https://file.aiquickdraw.com/mathbot/file/17254125135092gkjc2pd.jpg

The equation given in the image is:

\[
\sqrt{x^3} \cdot \sqrt{x} = x
\]

Let's solve this step by step.

1. First, simplify the square roots on the left-hand side. The square root of a product is the product of the square roots:

\[
\sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x \cdot \sqrt{x}
\]

So, the equation becomes:

\[
x \cdot \sqrt{x} \cdot \sqrt{x} = x
\]

2. Next, notice that \(\sqrt{x} \cdot \sqrt{x} = x\):

\[
x \cdot x = x
\]

This simplifies to:

\[
x^2 = x
\]

3. Now, to solve the equation \(x^2 = x\), subtract \(x\) from both sides:

\[
x^2 - x = 0
\]

4. Factor the equation:

\[
x(x - 1) = 0
\]

5. This gives us two possible solutions:

\[
x = 0 \quad \text{or} \quad x = 1
\]

Thus, the solutions are \(x = 0\) and \(x = 1\).

Would you like further details or have any questions?

Here are 5 questions related to this problem:
1. Why does \(\sqrt{x} \cdot \sqrt{x} = x\) instead of \(2x\)?
2. What does it mean when we factor an equation like \(x(x-1) = 0\)?
3. How does the value of \(x\) affect the original equation when \(x = 0\) or \(x = 1\)?
4. Can this method be used to solve other similar equations with square roots and exponents?
5. What are the implications of \(x = 0\) or \(x = 1\) in other algebraic contexts?

**Tip:** Always double-check if the solutions satisfy the original equation by substituting them back in!

Learn how to solve the algebraic equation involving square roots and exponents: √(x³) * √(x) = x. This step-by-step solution covers key concepts like square roots, exponents, and factoring quadratic equations, making it suitable for students in Grades 8-10.

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