1) $$ \frac{4^{x-1}+4^{x+1}}{17 \cdot 12^{x}} $$ Simplify 2) $$ \frac{\frac{3^{x+1}-3^{x-1}}{2 \cdot 3^{x}}}{ ext { 3) } \frac{5^{a-2} \cdot 2^{a+2}}{10^{a}-10^{a-1} \cdot 2}} \begin{array}{l} ext { 4) } \frac{3^{m+4}-6 \cdot 3^{m+1}}{7 \cdot 3^{m+2}}\end{array} $$

Question

1) $$ \frac{4^{x-1}+4^{x+1}}{17 \cdot 12^{x}} $$ Simplify 2) $$ \frac{\frac{3^{x+1}-3^{x-1}}{2 \cdot 3^{x}}}{	ext { 3) } \frac{5^{a-2} \cdot 2^{a+2}}{10^{a}-10^{a-1} \cdot 2}} \begin{array}{l}	ext { 4) } \frac{3^{m+4}-6 \cdot 3^{m+1}}{7 \cdot 3^{m+2}}\end{array} $$

UpStudy AI · Accepted Answer

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

---

### **Problem 1**

Simplify:
$$
\frac{4^{x-1} + 4^{x+1}}{17 \cdot 12^{x}}
$$

**Step 1: Factor the Numerator**

$$
4^{x-1} + 4^{x+1} = 4^{x-1} \left(1 + 4^{2}\right) = 4^{x-1} \cdot 17
$$

**Step 2: Substitute Back into the Expression**

$$
\frac{4^{x-1} \cdot 17}{17 \cdot 12^{x}} = \frac{4^{x-1}}{12^{x}}
$$

**Step 3: Simplify the Fraction**

Express $$4$$ and $$12$$ with base $$2$$ and $$3$$:

$$
\frac{4^{x-1}}{12^{x}} = \frac{(2^2)^{x-1}}{(2^2 \cdot 3)^x} = \frac{2^{2x - 2}}{2^{2x} \cdot 3^{x}} = \frac{1}{4 \cdot 3^{x}}
$$

**Final Simplified Form:**

$$
\frac{1}{4 \cdot 3^{x}}
$$

---

### **Problem 2**

Simplify:
$$
\frac{3^{x+1} - 3^{x-1}}{2 \cdot 3^{x}}
$$

**Step 1: Factor the Numerator**

$$
3^{x+1} - 3^{x-1} = 3^{x-1} (3^{2} - 1) = 3^{x-1} \cdot 8
$$

**Step 2: Substitute Back into the Expression**

$$
\frac{8 \cdot 3^{x-1}}{2 \cdot 3^{x}} = \frac{8}{2} \cdot \frac{3^{x-1}}{3^{x}} = 4 \cdot 3^{-1} = \frac{4}{3}
$$

**Final Simplified Form:**

$$
\frac{4}{3}
$$

---

### **Problem 3**

Simplify:
$$
\frac{5^{a-2} \cdot 2^{a+2}}{10^{a} - 10^{a-1} \cdot 2}
$$

**Step 1: Simplify the Denominator**

$$
10^{a} - 10^{a-1} \cdot 2 = 10^{a-1} (10 - 2) = 10^{a-1} \cdot 8
$$

**Step 2: Express $$10^{a-1}$$ in Terms of Base $$2$$ and $$5$$**

$$
10^{a-1} = (2 \cdot 5)^{a-1} = 2^{a-1} \cdot 5^{a-1}
$$

**Step 3: Substitute Back into the Expression**

$$
\frac{5^{a-2} \cdot 2^{a+2}}{8 \cdot 2^{a-1} \cdot 5^{a-1}}}
$$

**Step 4: Simplify the Fraction**

$$
\frac{2^{a+2}}{2^{a-1}} = 2^{3} = 8
$$
$$
\frac{5^{a-2}}{5^{a-1}} = 5^{-1} = \frac{1}{5}
$$
$$
\frac{8 \cdot \frac{1}{5}}{8} = \frac{1}{5}
$$

**Final Simplified Form:**

$$
\frac{1}{5}
$$

---

### **Problem 4**

Simplify:
$$
\frac{3^{m+4} - 6 \cdot 3^{m+1}}{7 \cdot 3^{m+2}}
$$

**Step 1: Factor the Numerator**

$$
3^{m+4} - 6 \cdot 3^{m+1} = 3^{m} \cdot 81 - 6 \cdot 3^{m} \cdot 3 = 3^{m} (81 - 18) = 3^{m} \cdot 63
$$

**Step 2: Substitute Back into the Expression**

$$
\frac{63 \cdot 3^{m}}{7 \cdot 3^{m+2}}}
$$

**Step 3: Simplify the Fraction**

$$
\frac{63}{7} = 9
$$
$$
\frac{3^{m}}{3^{m+2}} = 3^{-2} = \frac{1}{9}
$$
$$
9 \cdot \frac{1}{9} = 1
$$

**Final Simplified Form:**

$$
1
$$

---

### **Summary of Simplified Expressions**

1. $$\displaystyle \frac{1}{4 \cdot 3^{x}}$$
2. $$\displaystyle \frac{4}{3}$$
3. $$\displaystyle \frac{1}{5}$$
4. $$1$$
1. $$ \frac{1}{4 \cdot 3^{x}} $$ 2. $$ \frac{4}{3} $$ 3. $$ \frac{1}{5} $$ 4. $$ 1 $$

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