If `4 le p le 5 and -10 le q le -9`, then the least value is given by the expression :

Options

A. `p.q`
B. `pq^6`
C. `p^6q`
D. (pq)^5`

Answer

To solve the problem, we need to find the least value of the expression given the constraints on \( p \) and \( q \).

### Step-by-Step Solution:

1. **Identify the Range of \( p \) and \( q \)**:
- We are given that \( 4 \leq p \leq 5 \) and \( -10 \leq q \leq -9 \).
- This means the minimum value of \( p \) is 4 and the maximum value of \( q \) is -10.

2. **Choose the Minimum Values**:
- To find the least value of the expression, we should take the minimum value of \( p \) and the minimum value of \( q \).
- Therefore, we take \( p = 4 \) and \( q = -10 \).

3. **Evaluate the Expressions**:
- We need to evaluate the expressions given in the options. Let's denote the expressions as follows:
- Option A: \( PQ \)
- Option B: \( PQ^6 \)
- Option C: \( P^6 Q \)
- Option D: \( P^Q \)

4. **Calculate Each Expression**:
- **Option A**:
\[ PQ = 4 \times (-10) = -40 \]
- **Option B**:
\[ PQ^6 = 4 \times (-10)^6 = 4 \times 1000000 = 4000000 \quad (\text{positive value}) \]
- **Option C**:
\[ P^6 Q = 4^6 \times (-10) = 4096 \times (-10) = -40960 \]
- **Option D**:
\[ P^Q = 4^{-10} = \frac{1}{4^{10}} \quad (\text{very small positive value}) \]

5. **Compare the Values**:
- Now we compare the calculated values:
- Option A: \( -40 \)
- Option B: \( 4000000 \)
- Option C: \( -40960 \)
- Option D: \( \text{very small positive value} \)

6. **Determine the Least Value**:
- The least value among these is \( -40960 \) from Option C.

### Conclusion:
The least value is given by the expression \( P^6 Q \), which corresponds to Option C.