Given: Quantity I = $18x^3 y^3$ Quantity II = $12x^4 y^4$ where $x 0$ and $y

Question

Given:

Quantity I = $18x^3 y^3$

Quantity II = $12x^4 y^4$

where $x > 0$ and $y < 0$.

Compare Quantity I and Quantity II.

Accepted Answer

**Solution:**

1. Given $x > 0$ and $y < 0$.

2. Quantity I = $18x^3 y^3$ and Quantity II = $12x^4 y^4$.

3. Since $x$ is positive, powers of $x$ remain positive.

4. However, $y$ is negative. 
   - $y^3 = y 	imes y 	imes y$ is negative because an odd power of a negative number is negative.
   - $y^4 = y 	imes y 	imes y 	imes y$ is positive because an even power of a negative number is positive.

5. Therefore:
   - Quantity I = positive constant $	imes$ positive $x^3$ $	imes$ negative $y^3$ = negative value.
   - Quantity II = positive constant $	imes$ positive $x^4$ $	imes$ positive $y^4$ = positive value.

6. Since Quantity I is negative and Quantity II is positive, Quantity I is always less than Quantity II.

**Answer:** Quantity I < Quantity II

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