In the given questions, two quantities are provided: Quantity I and Quantity II. Determine the relationship between these two quantities and select the correct option.

**Quantity I:** Value of *x* satisfying the equation $\left(\sqrt{p} imes x ight)^2 = p^4 imes \sqrt{p^{16}}$

**Quantity II:** Value of *y* satisfying the equation $\sqrt{y^4} = \sqrt{256}$

Question

In the given questions, two quantities are provided: Quantity I and Quantity II. Determine the relationship between these two quantities and select the correct option.

**Quantity I:** Value of *x* satisfying the equation $\left(\sqrt{p} 	imes xight)^2 = p^4 	imes \sqrt{p^{16}}$

**Quantity II:** Value of *y* satisfying the equation $\sqrt{y^4} = \sqrt{256}$

Accepted Answer

**Solution:**

1. **Evaluate Quantity I:**
   Given:
   $$\left(\sqrt{p} 	imes xight)^2 = p^4 	imes \sqrt{p^{16}}$$

Simplify the right side:
   - $\sqrt{p^{16}} = p^{16/2} = p^8$

So the equation becomes:
   $$\left(\sqrt{p} 	imes xight)^2 = p^4 	imes p^8 = p^{12}$$

The left side:
   $$\left(\sqrt{p} 	imes xight)^2 = (p^{1/2} 	imes x)^2 = p^{1} 	imes x^2$$

Therefore:
   $$p^{1} 	imes x^2 = p^{12}$$

Divide both sides by $p^{1}$:
   $$x^2 = p^{11}$$

Taking square root:
   $$x = p^{11/2}$$

2. **Evaluate Quantity II:**
   Given:
   $$\sqrt{y^4} = \sqrt{256}$$

Simplify both sides:
   - Left side: $\sqrt{y^4} = y^{4/2} = y^2$
   - Right side: $\sqrt{256} = 16$

So:
   $$y^2 = 16$$

Taking square root:
   $$y = 4$$

3. **Compare Quantity I and Quantity II:**
   - Quantity I = $p^{11/2}$
   - Quantity II = 4

Since $p$ is a variable and no value or range is given, the relationship between $p^{11/2}$ and 4 cannot be determined definitively.

**Answer:** Quantity I = Quantity II or no relation

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