(9) $$ -s^{3}-5 s t $$ when $$ s=-3 $$ and $$ t=-2 $$ (10) $$ \frac{a^{2}-b^{2}}{a-b} $$ when $$ a=5 $$ and $$ b=-9 $$ when $$ g=2, h=-3 $$, and $$ w=-7 $$ $$ g^{2}-3(h-g)-w^{2} $$ (12) $$ r^{3}-\left[q+(r-s)^{2}\right] $$ when $$ r=3, q=2 $$, and $$ s=5 $$

Question

UpStudy AI · Accepted Answer

Sure, let's evaluate each of the given expressions with the specified values step by step.

---

### Problem (9)
$$ -s^{3} - 5 s t $$
**Given:** $$ s = -3 $$, $$ t = -2 $$

**Step-by-Step Evaluation:**

1. **Calculate $$ s^{3} $$:**
   $$
   s^{3} = (-3)^{3} = -27
   $$
   
2. **Calculate $$ 5 s t $$:**
   $$
   5 s t = 5 \times (-3) \times (-2) = 5 \times 6 = 30
   $$
   
3. **Combine the results:**
   $$
   -s^{3} - 5 s t = -(-27) - 30 = 27 - 30 = -3
   $$

**Answer:** $$-3$$

---

### Problem (10)
$$ \frac{a^{2} - b^{2}}{a - b} $$
**Given:** $$ a = 5 $$, $$ b = -9 $$

**Step-by-Step Evaluation:**

1. **Calculate $$ a^{2} $$ and $$ b^{2} $$:**
   $$
   a^{2} = 5^{2} = 25
   $$
   $$
   b^{2} = (-9)^{2} = 81
   $$
   
2. **Compute the numerator $$ a^{2} - b^{2} $$:**
   $$
   a^{2} - b^{2} = 25 - 81 = -56
   $$
   
3. **Compute the denominator $$ a - b $$:**
   $$
   a - b = 5 - (-9) = 5 + 9 = 14
   $$
   
4. **Divide the numerator by the denominator:**
   $$
   \frac{a^{2} - b^{2}}{a - b} = \frac{-56}{14} = -4
   $$

**Answer:** $$-4$$

---

### Problem (11)
$$ g^{2} - 3(h - g) - w^{2} $$
**Given:** $$ g = 2 $$, $$ h = -3 $$, $$ w = -7 $$

**Step-by-Step Evaluation:**

1. **Calculate $$ g^{2} $$ and $$ w^{2} $$:**
   $$
   g^{2} = 2^{2} = 4
   $$
   $$
   w^{2} = (-7)^{2} = 49
   $$
   
2. **Calculate $$ h - g $$:**
   $$
   h - g = -3 - 2 = -5
   $$
   
3. **Multiply by 3:**
   $$
   3(h - g) = 3 \times (-5) = -15
   $$
   
4. **Combine all parts:**
   $$
   g^{2} - 3(h - g) - w^{2} = 4 - (-15) - 49 = 4 + 15 - 49 = 19 - 49 = -30
   $$

**Answer:** $$-30$$

---

### Problem (12)
$$ r^{3} - \left[ q + (r - s)^{2} \right] $$
**Given:** $$ r = 3 $$, $$ q = 2 $$, $$ s = 5 $$

**Step-by-Step Evaluation:**

1. **Calculate $$ r^{3} $$:**
   $$
   r^{3} = 3^{3} = 27
   $$
   
2. **Calculate $$ r - s $$:**
   $$
   r - s = 3 - 5 = -2
   $$
   
3. **Calculate $$ (r - s)^{2} $$:**
   $$
   (r - s)^{2} = (-2)^{2} = 4
   $$
   
4. **Compute the expression inside the brackets:**
   $$
   q + (r - s)^{2} = 2 + 4 = 6
   $$
   
5. **Subtract the bracketed term from $$ r^{3} $$:**
   $$
   r^{3} - [q + (r - s)^{2}] = 27 - 6 = 21
   $$

**Answer:** $$21$$

---

### Summary of Answers
1. **Problem (9):** $$-3$$
2. **Problem (10):** $$-4$$
3. **Problem (11):** $$-30$$
4. **Problem (12):** $$21$$

If you have any further questions or need more explanations, feel free to ask!
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