https://file.aiquickdraw.com/mathbot/file/17276190264877f4zkwlj.jpg

The problem in the image asks you to find the numerical values of two algebraic expressions for given values of \( x \) and \( y \). Here are the two expressions with their corresponding conditions:

1. **Part (a):**
   \[
   -x^5 + 2x^3y^2
   \]
   for \( x = -3 \) and \( y = -1 \).

2. **Part (b):**
   \[
   4x^3y + xy^2 - 1
   \]
   for \( x = -\frac{1}{2} \) and \( y = -2 \).

---

Let's solve each part step by step.

### Part (a) Solution:
The expression is:
\[
-x^5 + 2x^3y^2
\]
Substitute \( x = -3 \) and \( y = -1 \):

\[
-x^5 = -(-3)^5 = -(-243) = 243
\]
Next, for the second term:
\[
2x^3y^2 = 2(-3)^3(-1)^2 = 2(-27)(1) = -54
\]
Now, adding both terms:
\[
243 + (-54) = 189
\]
So, the result for part (a) is **189**.

### Part (b) Solution:
The expression is:
\[
4x^3y + xy^2 - 1
\]
Substitute \( x = -\frac{1}{2} \) and \( y = -2 \):

First term:
\[
4x^3y = 4\left(-\frac{1}{2}\right)^3(-2) = 4\left(-\frac{1}{8}\right)(-2) = 4 \times \frac{1}{4} = 1
\]
Second term:
\[
xy^2 = \left(-\frac{1}{2}\right)(-2)^2 = \left(-\frac{1}{2}\right)(4) = -2
\]
Now, combine all terms:
\[
1 + (-2) - 1 = -2
\]
So, the result for part (b) is **-2**.

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Let me know if you'd like more details or have any questions!  
Here are five related questions to explore further:

1. How do you simplify expressions with negative exponents?
2. What happens when a negative number is raised to an odd versus an even power?
3. How do you handle operations involving fractions and powers?
4. How can algebraic expressions be factored to simplify computation?
5. How does multiplying by negative values affect the result of an expression?

**Tip:** Always be careful with signs when working with negative values, especially when they are raised to powers!

Find the numerical values of the following algebraic expressions for given values of x and y:
1. -x^5 + 2x^3y^2 for x = -3 and y = -1
2. 4x^3y + xy^2 - 1 for x = -1/2 and y = -2

Learn how to find the numerical values of algebraic expressions for specified values of variables x and y. This problem covers polynomial expressions, substitution, and basic arithmetic operations, making it suitable for Grade 9 students.

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