1. $$ \frac{3 a^{3} imes 5 a^{5}}{\left(5 a^{3} ight)^{2}} $$ 2. $$ -2\left(-4 a^{2}+b^{3} ight)^{2} $$ 3. $$ 6 a^{4}+9 a^{4} $$ 4. $$ p^{x-1} \cdot p^{x+1} $$ 5 $$ 16^{4 x} \div 4 $$

Question

1. $$ \frac{3 a^{3} 	imes 5 a^{5}}{\left(5 a^{3}ight)^{2}} $$ 2. $$ -2\left(-4 a^{2}+b^{3}ight)^{2} $$ 3. $$ 6 a^{4}+9 a^{4} $$ 4. $$ p^{x-1} \cdot p^{x+1} $$ 5 $$ 16^{4 x} \div 4 $$

UpStudy AI · Accepted Answer

Let's simplify each of the given expressions step by step.

---

### **1. $$ \frac{3 a^{3} \times 5 a^{5}}{\left(5 a^{3}\right)^{2}} $$**

**Step 1: Simplify the numerator**

$$
3 a^{3} \times 5 a^{5} = (3 \times 5) \times a^{3+5} = 15 a^{8}
$$

**Step 2: Simplify the denominator**

$$
\left(5 a^{3}\right)^{2} = 5^{2} \times (a^{3})^{2} = 25 a^{6}
$$

**Step 3: Combine numerator and denominator**

$$
\frac{15 a^{8}}{25 a^{6}} = \frac{15}{25} \times a^{8-6} = \frac{3}{5} a^{2}
$$

**Final Answer:**  
$$
\frac{3}{5} a^{2}
$$

---

### **2. $$ -2\left(-4 a^{2}+b^{3}\right)^{2} $$**

**Step 1: Expand the squared term**

$$
\left(-4 a^{2} + b^{3}\right)^{2} = (-4 a^{2})^{2} + 2(-4 a^{2})(b^{3}) + (b^{3})^{2} = 16 a^{4} - 8 a^{2} b^{3} + b^{6}
$$

**Step 2: Distribute the $$-2$$ across the expanded terms**

$$
-2 \times (16 a^{4} - 8 a^{2} b^{3} + b^{6}) = -32 a^{4} + 16 a^{2} b^{3} - 2 b^{6}
$$

**Final Answer:**  
$$
-32 a^{4} + 16 a^{2} b^{3} - 2 b^{6}
$$

---

### **3. $$ 6 a^{4}+9 a^{4} $$**

**Combine like terms:**

$$
6 a^{4} + 9 a^{4} = (6 + 9) a^{4} = 15 a^{4}
$$

**Final Answer:**  
$$
15 a^{4}
$$

---

### **4. $$ p^{x-1} \cdot p^{x+1} $$**

**Use the law of exponents $$ p^{m} \cdot p^{n} = p^{m+n} $$:**

$$
p^{x-1} \cdot p^{x+1} = p^{(x-1)+(x+1)} = p^{2x}
$$

**Final Answer:**  
$$
p^{2x}
$$

---

### **5. $$ 16^{4x} \div 4 $$**

**Step 1: Express $$16$$ and $$4$$ as powers of $$4$$:**

$$
16 = 4^{2} \quad \text{and} \quad 4 = 4^{1}
$$

**Step 2: Rewrite the expression using these bases:**

$$
16^{4x} \div 4 = (4^{2})^{4x} \div 4^{1} = 4^{8x} \div 4^{1}
$$

**Step 3: Subtract exponents when dividing like bases:**

$$
4^{8x} \div 4^{1} = 4^{8x - 1}
$$

**Final Answer:**  
$$
4^{8x - 1}
$$

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Feel free to reach out if you need further clarification on any of these steps!
1. $$ \frac{3}{5} a^{2} $$ 2. $$ -32 a^{4} + 16 a^{2} b^{3} - 2 b^{6} $$ 3. $$ 15 a^{4} $$ 4. $$ p^{2x} $$ 5. $$ 4^{8x - 1} $$

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