Let's break down the math problem from the image you provided:

1. Simplifying $x^3 \cdot x^2$

The expression is $x^3 \cdot x^2$. Using the product of powers rule, when you multiply two powers with the same base, you add the exponents.

Thus, $x^3 \cdot x^2 = x^{3+2} = x^5$

This simplification follows the exponent rule $x^a \cdot x^b = x^{a+b}$.

2. Question 2: Filling the blanks

For the expression given: $x^3 \cdot x^2 = (x \cdot x \cdot x) \cdot (\_\_\_\_ \cdot \_\_\_\_)$ You are asked to fill in the blanks. The expanded form of $x^2$ is $x \cdot x$.

So, the complete expanded form of $x^3 \cdot x^2$ becomes: $x^3 \cdot x^2 = (x \cdot x \cdot x) \cdot (x \cdot x)$

3. Question 3: What did you do with the exponents?

In the problem $x^3 \cdot x^2$, you added the exponents 3 and 2 to get the final exponent 5. The rule applied is: $x^3 \cdot x^2 = x^{3+2} = x^5$

Would you like any further clarification or additional details on these rules?

Related questions:

1. What is the product rule for exponents, and why does it work?
2. Can the product rule for exponents be applied to numbers with different bases?
3. How would you simplify $y^4 \cdot y^5$?
4. What happens if one of the exponents is zero, such as $x^0 \cdot x^4$?
5. How does the product of powers rule apply to negative exponents?

Tip:

When simplifying expressions with exponents, always ensure the base is the same before applying the rule of adding exponents!