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) $$ \frac{27^{x+1} .18^{x-1}}{36^{3-x}} $$ j) $$ \frac{54^{x+1} .36^{x-1}}{24^{2 x-3}} $$

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) $$ \frac{27^{x+1} .18^{x-1}}{36^{3-x}} $$ j) $$ \frac{54^{x+1} .36^{x-1}}{24^{2 x-3}} $$

UpStudy AI · Accepted Answer

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

---

### **f) Simplify the expression:**

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

**Step 1:** Simplify each part individually.

1. **Simplify the negative exponent:**
   
   $$
   \left(\frac{h^{4}}{f^{2} g}\right)^{-2} = \left(\frac{f^{2} g}{h^{4}}\right)^{2} = \frac{f^{4} g^{2}}{h^{8}}
   $$

2. **Simplify $$ a^{0} $$:**
   
   $$
   a^{0} = 1
   $$

**Step 2:** Substitute the simplified parts back into the original expression.

$$
\frac{f g}{h^{3}} \times \frac{f^{4} g^{2}}{h^{8}} \div \frac{f^{2} g^{-2}}{h^{-3}}
$$

**Step 3:** Perform the multiplication.

$$
\frac{f g \cdot f^{4} g^{2}}{h^{3} \cdot h^{8}} = \frac{f^{5} g^{3}}{h^{11}}
$$

**Step 4:** Perform the division by multiplying by the reciprocal.

$$
\frac{f^{5} g^{3}}{h^{11}} \times \frac{h^{-3}}{f^{2} g^{-2}} = \frac{f^{5} g^{3} \cdot h^{-3}}{h^{11} \cdot f^{2} g^{-2}}
$$

**Step 5:** Simplify by combining like terms using the law $$ a^{m} \cdot a^{n} = a^{m+n} $$ and $$ \frac{a^{m}}{a^{n}} = a^{m-n} $$.

$$
\frac{f^{5} \cdot h^{-3}}{f^{2}} \times \frac{g^{3}}{g^{-2} \cdot h^{11}} = f^{5-2} \cdot g^{3 - (-2)} \cdot h^{-3 -11} = f^{3} g^{5} h^{-14}
$$

**Step 6:** Express negative exponents as reciprocals.

$$
f^{3} g^{5} \cdot \frac{1}{h^{14}} = \frac{f^{3} g^{5}}{h^{14}}
$$

**Final Simplified Form:**

$$
\boxed{\frac{f^{3} g^{5}}{h^{14}}}
$$

---

### **h) Simplify the expression:**

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

**Step 1:** Express each number as a power of its prime factors.

- $$27 = 3^{3}$$
- $$18 = 2 \cdot 3^{2}$$
- $$36 = 2^{2} \cdot 3^{2}$$

**Step 2:** Rewrite the expression using prime factors.

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

**Step 3:** Apply the power rule $$ (a^{m})^{n} = a^{mn} $$.

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

**Step 4:** Simplify the exponents.

$$
\frac{3^{3x + 3} \cdot 2^{x - 1} \cdot 3^{2x - 2}}{2^{6 - 2x} \cdot 3^{6 - 2x}}
$$

**Step 5:** Combine like bases by adding exponents in the numerator and keeping denominators as they are.

- For base 3:

$$
3^{3x + 3} \cdot 3^{2x - 2} = 3^{5x + 1}
$$

- For base 2:

$$
2^{x - 1}
$$

**Step 6:** Rewrite the entire expression.

$$
\frac{3^{5x + 1} \cdot 2^{x - 1}}{2^{6 - 2x} \cdot 3^{6 - 2x}}
$$

**Step 7:** Use the quotient rule $$ \frac{a^{m}}{a^{n}} = a^{m-n} $$.

- For base 3:

$$
3^{5x + 1 - (6 - 2x)} = 3^{5x + 1 - 6 + 2x} = 3^{7x - 5}
$$

- For base 2:

$$
2^{x - 1 - (6 - 2x)} = 2^{x - 1 - 6 + 2x} = 2^{3x - 7}
$$

**Final Simplified Form:**

$$
\boxed{2^{3x - 7} \cdot 3^{7x - 5}}
$$

---

### **j) Simplify the expression:**

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

**Step 1:** Express each number as a power of its prime factors.

- $$54 = 2 \cdot 3^{3}$$
- $$36 = 2^{2} \cdot 3^{2}$$
- $$24 = 2^{3} \cdot 3$$

**Step 2:** Rewrite the expression using prime factors.

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

**Step 3:** Apply the power rule $$ (a^{m} \cdot b^{n})^{p} = a^{mp} \cdot b^{np} $$.

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

**Step 4:** Simplify the exponents.

$$
\frac{2^{x +1} \cdot 3^{3x +3} \cdot 2^{2x -2} \cdot 3^{2x -2}}{2^{6x -9} \cdot 3^{2x -3}}
$$

**Step 5:** Combine like bases in the numerator.

- For base 2:

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

- For base 3:

$$
3^{3x +3} \cdot 3^{2x -2} = 3^{5x +1}
$$

**Step 6:** Rewrite the entire expression.

$$
\frac{2^{3x -1} \cdot 3^{5x +1}}{2^{6x -9} \cdot 3^{2x -3}}
$$

**Step 7:** Use the quotient rule $$ \frac{a^{m}}{a^{n}} = a^{m-n} $$.

- For base 2:

$$
2^{3x -1 - (6x -9)} = 2^{3x -1 -6x +9} = 2^{-3x +8}
$$

- For base 3:

$$
3^{5x +1 - (2x -3)} = 3^{5x +1 -2x +3} = 3^{3x +4}
$$

**Step 8:** Express negative exponents as reciprocals if desired, but it's acceptable to leave exponents as is.

**Final Simplified Form:**

$$
\boxed{2^{-3x +8} \cdot 3^{3x +4}}
$$

Alternatively, to write with positive exponents for base 2:

$$
\boxed{\frac{3^{3x +4}}{2^{3x -8}}}
$$

---

These simplifications utilize the fundamental laws of exponents, including the product of powers, power of a power, and the quotient of powers. Always ensure to factor numbers into their prime bases to effectively apply these exponent rules.
**f) Simplified Expression:** $$ \frac{f^{3} g^{5}}{h^{14}} $$ **h) Simplified Expression:** $$ 2^{3x -7} \cdot 3^{7x -5} $$ **j) Simplified Expression:** $$ 2^{-3x +8} \cdot 3^{3x +4} $$

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