Simplify radical expressions and perform number theory operations involving primes and composites.

$$\sqrt{8x^2y} = \sqrt{4 \cdot 2 \cdot x^2 \cdot y} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{2y} = 2x\sqrt{2y}$$

The simplified expression is:

$$\boxed{2x\sqrt{2y}}$$

2. Step 1: Identify prime numbers between 0 and 10. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. The prime numbers between 0 and 10 are 2, 3, 5, 7. Writing them in ascending order forms number $A$:

$$A = 2357$$

Odd numbers are integers not divisible by 2.
The odd numbers between 0 and 10 are 1, 3, 5, 7, 9.
Writing them in descending order forms number $B$:

$$B = 97531$$

$$A+B = 2357 + 97531 = 99888$$

The value of $A+B$ is:

$$\boxed{99888}$$

3. Step 1: List the numbers between 18 and 26. The numbers are 19, 20, 21, 22, 23, 24, 25.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
A composite number is a natural number greater than 1 that is not prime (i.e., it has at least one divisor other than 1 and itself).

• 19: Its only divisors are 1 and 19. So, 19 is a prime number.
• 20: It is divisible by 1, 2, 4, 5, 10, 20. So, 20 is a composite number.
• 21: It is divisible by 1, 3, 7, 21. So, 21 is a composite number.
• 22: It is divisible by 1, 2, 11, 22. So, 22 is a composite number.
• 23: Its only divisors are 1 and 23. So, 23 is a prime number.
• 24: It is divisible by 1, 2, 3, 4, 6, 8, 12, 24. So, 24 is a composite number.
• 25: It is divisible by 1, 5, 25. So, 25 is a composite number.

The classification is:

• Prime numbers: 19, 23
• Composite numbers: 20, 21, 22, 24, 25