f) $$ \frac{f g}{h^{3}} imes\left(\frac{h^{4}}{f^{2} g} ight)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}} $$ h) $$ \frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}} $$ j) $$ \frac{54^{x+1} \cdot 36^{x-1}}{24^{2 x-3}} $$

Question

f) $$ \frac{f g}{h^{3}} 	imes\left(\frac{h^{4}}{f^{2} g}ight)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}} $$ h) $$ \frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}} $$ j) $$ \frac{54^{x+1} \cdot 36^{x-1}}{24^{2 x-3}} $$

UpStudy AI · Accepted Answer

Let's simplify each of the given expressions using the rules of exponents.

---

### **Problem f)**

$$
\frac{f g}{h^{3}} \times \left(\frac{h^{4}}{f^{2} g}\right)^{-2} \div \frac{a^{0} f^{2} g^{-2}}{h^{-3}}
$$

**Step-by-Step Simplification:**

1. **Simplify $$a^0$$:**
   $$
   a^0 = 1
   $$
   
2. **Apply the negative exponent:**
   $$
   \left(\frac{h^4}{f^2 g}\right)^{-2} = \left(\frac{f^2 g}{h^4}\right)^2 = \frac{f^4 g^2}{h^8}
   $$
   
3. **Rewrite the division as multiplication by the reciprocal:**
   $$
   \div \frac{f^2 g^{-2}}{h^{-3}} = \times \frac{h^{-3}}{f^2 g^{-2}} = h^{-3} \times f^{-2} \times g^{2}
   $$
   
4. **Combine all parts:**
   $$
   \frac{f g}{h^{3}} \times \frac{f^4 g^2}{h^8} \times f^{-2} g^{2} h^{-3}
   $$
   
5. **Combine like terms using exponent addition:**
   $$
   f^{1 + 4 - 2} \times g^{1 + 2 + 2} \times h^{-3 - 8 - 3} = f^3 g^5 h^{-14}
   $$
   
6. **Express with positive exponents:**
   $$
   \frac{f^3 g^5}{h^{14}}
   $$

**Final Simplified Form:**
$$
\boxed{\dfrac{f^{3} g^{5}}{h^{14}}}
$$

---

### **Problem h)**

$$
\frac{27^{x+1} \cdot 18^{x-1}}{36^{3-x}}
$$

**Step-by-Step Simplification:**

1. **Express each base as powers of prime numbers (2 and 3):**
   $$
   27 = 3^3, \quad 18 = 2 \times 3^2, \quad 36 = 2^2 \times 3^2
   $$
   
2. **Rewrite the expression with these bases:**
   $$
   \frac{(3^3)^{x+1} \cdot (2 \times 3^2)^{x-1}}{(2^2 \times 3^2)^{3-x}}
   $$
   
3. **Apply the power of a power rule:**
   $$
   \frac{3^{3(x+1)} \cdot 2^{x-1} \cdot 3^{2(x-1)}}{2^{2(3-x)} \cdot 3^{2(3-x)}}
   $$
   
4. **Simplify the exponents:**
   $$
   \frac{3^{3x+3} \cdot 2^{x-1} \cdot 3^{2x-2}}{2^{6-2x} \cdot 3^{6-2x}} = \frac{2^{x-1} \cdot 3^{5x+1}}{2^{6-2x} \cdot 3^{6-2x}}
   $$
   
5. **Subtract exponents in the numerator and denominator:**
   $$
   2^{x-1 - (6-2x)} \cdot 3^{5x+1 - (6-2x)} = 2^{3x-7} \cdot 3^{7x-5}
   $$

**Final Simplified Form:**
$$
\boxed{2^{3x - 7} \cdot 3^{7x - 5}}
$$

---

### **Problem j)**

$$
\frac{54^{x+1} \cdot 36^{x-1}}{24^{2x-3}}
$$

**Step-by-Step Simplification:**

1. **Express each base as powers of prime numbers (2 and 3):**
   $$
   54 = 2 \times 3^3, \quad 36 = 2^2 \times 3^2, \quad 24 = 2^3 \times 3
   $$
   
2. **Rewrite the expression with these bases:**
   $$
   \frac{(2 \times 3^3)^{x+1} \cdot (2^2 \times 3^2)^{x-1}}{(2^3 \times 3)^{2x-3}}
   $$
   
3. **Apply the power of a power rule:**
   $$
   \frac{2^{x+1} \cdot 3^{3(x+1)} \cdot 2^{2(x-1)} \cdot 3^{2(x-1)}}{2^{3(2x-3)} \cdot 3^{2x-3}}
   $$
   
4. **Simplify the exponents:**
   $$
   \frac{2^{x+1 + 2x-2} \cdot 3^{3x+3 + 2x-2}}{2^{6x-9} \cdot 3^{2x-3}} = \frac{2^{3x-1} \cdot 3^{5x+1}}{2^{6x-9} \cdot 3^{2x-3}}
   $$
   
5. **Subtract exponents in the numerator and denominator:**
   $$
   2^{3x-1 - (6x-9)} \cdot 3^{5x+1 - (2x-3)} = 2^{-3x +8} \cdot 3^{3x +4}
   $$
   
6. **Express with positive exponents (optional):**
   $$
   3^{3x +4} \cdot 2^{8 -3x}
   $$

**Final Simplified Form:**
$$
\boxed{3^{3x + 4} \cdot 2^{8 - 3x}}
$$

---

These simplified forms utilize the fundamental properties of exponents to condense the original expressions into more manageable forms.
**Simplified Answers:** f) $$ \frac{f^{3} g^{5}}{h^{14}} $$ h) $$ 2^{3x - 7} \cdot 3^{7x - 5} $$ j) $$ 3^{3x + 4} \cdot 2^{8 - 3x} $$

f)
h)
j)

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f) h) j)

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f)
h)
j)