**Given:** If *a*, *b*, *c*, and *d* are positive integers, compare the following quantities: (i) $$\frac{a^{-b}}{a^{-a}} = ab imes c$$ (ii) $$\frac{a^3 imes b^3}{ab^2} = \frac{d^4 imes b^3}{d imes b}$$ **Quantity I:** Find the value of *c*. **Quantity II:** Find the value of *d*. **Compare Quantity I and Quantity II:** (a) Quantity I Quantity II (b) Quantity I

Question

**Given:**

If *a*, *b*, *c*, and *d* are positive integers, compare the following quantities:

(i) $$\frac{a^{-b}}{a^{-a}} = ab 	imes c$$

(ii) $$\frac{a^3 	imes b^3}{ab^2} = \frac{d^4 	imes b^3}{d 	imes b}$$

**Quantity I:** Find the value of *c*.

**Quantity II:** Find the value of *d*.

**Compare Quantity I and Quantity II:**

(a) Quantity I > Quantity II

(b) Quantity I < Quantity II

(c) Quantity I \geq Quantity II

(d) Quantity I \leq Quantity II

(e) Quantity I = Quantity II or no relation

Accepted Answer

**Solution:**

1. Analyze Quantity I:

Given:
$$\frac{a^{-b}}{a^{-a}} = ab 	imes c$$

Using the laws of exponents:

$$\frac{a^{-b}}{a^{-a}} = a^{-b - (-a)} = a^{-b + a} = a^{a - b}$$

So,

$$a^{a - b} = ab 	imes c$$

Since *a* and *b* are positive integers, the left side is a power of *a*, and the right side is the product of *a*, *b*, and *c*.

To find *c*, express the right side as:

$$ab 	imes c = a^{a - b}$$

Divide both sides by *ab*:

$$c = \frac{a^{a - b}}{ab}$$

2. Analyze Quantity II:

Given:

$$\frac{a^3 	imes b^3}{ab^2} = \frac{d^4 	imes b^3}{d 	imes b}$$

Simplify left side:

$$\frac{a^3 b^3}{a b^2} = a^{3-1} b^{3-2} = a^2 b^1 = a^2 b$$

Simplify right side:

$$\frac{d^4 b^3}{d b} = d^{4-1} b^{3-1} = d^3 b^2$$

Equate both sides:

$$a^2 b = d^3 b^2$$

Divide both sides by *b*:

$$a^2 = d^3 b$$

Since *a*, *b*, and *d* are positive integers, to find *d*:

$$d^3 = \frac{a^2}{b}$$

3. Comparing *c* and *d*:

- From Quantity I, *c* depends on the expression $$\frac{a^{a - b}}{ab}$$.
- From Quantity II, *d* depends on $$\sqrt[3]{\frac{a^2}{b}}$$.

Given the variables are positive integers, and since the powers and bases differ, generally, the value of *c* will be less than *d*.

**Answer:** Quantity I < Quantity II

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