Classify the following numbers as rational or irrational number:
(i) `sqrt(27)` (ii) `sqrt(36` (iii) `sqrt(5) xx sqrt(125)` (iv) `2 sqrt(3)`
(v) `(7 sqrt(7))/(sqrt(343))` (vi) `2 + sqrt(21)` (viii) `5 + 2 sqrt(23) - (sqrt(25) + sqrt(92))`
(viii) `(22)/(7)` (ix) `pi` (x) `.^(3)sqrt(37)`

Answer

To classify the given numbers as rational or irrational, we will analyze each number step by step.

### Step-by-Step Solution:

**(i) `sqrt(27)`**
- We can simplify `sqrt(27)` as follows:
`sqrt(27) = sqrt(9 * 3) = sqrt(9) * sqrt(3) = 3 * sqrt(3)`
- Since `sqrt(3)` is an irrational number, and multiplying it by 3 (a rational number) results in an irrational number.
**Conclusion:** `sqrt(27)` is **irrational**.

**(ii) `sqrt(36)`**
- We can simplify `sqrt(36)` as follows:
`sqrt(36) = 6`
- Since 6 is a rational number,
**Conclusion:** `sqrt(36)` is **rational**.

**(iii) `sqrt(5) * sqrt(125)`**
- We can simplify `sqrt(125)` as follows:
`sqrt(125) = sqrt(25 * 5) = sqrt(25) * sqrt(5) = 5 * sqrt(5)`
- Therefore, `sqrt(5) * sqrt(125) = sqrt(5) * (5 * sqrt(5)) = 5 * (sqrt(5) * sqrt(5)) = 5 * 5 = 25`
- Since 25 is a rational number,
**Conclusion:** `sqrt(5) * sqrt(125)` is **rational**.

**(iv) `2 * sqrt(3)`**
- Here, 2 is a rational number and `sqrt(3)` is an irrational number.
- The product of a rational number and an irrational number is irrational.
**Conclusion:** `2 * sqrt(3)` is **irrational**.

**(v) `(7 * sqrt(7)) / (sqrt(343))`**
- We can simplify `sqrt(343)` as follows:
`sqrt(343) = sqrt(7 * 49) = sqrt(7) * sqrt(49) = sqrt(7) * 7`
- Therefore, `(7 * sqrt(7)) / (sqrt(343)) = (7 * sqrt(7)) / (7 * sqrt(7)) = 1`
- Since 1 is a rational number,
**Conclusion:** `(7 * sqrt(7)) / (sqrt(343))` is **rational**.

**(vi) `2 + sqrt(21)`**
- Here, 2 is a rational number and `sqrt(21)` is an irrational number.
- The sum of a rational number and an irrational number is irrational.
**Conclusion:** `2 + sqrt(21)` is **irrational**.

**(vii) `5 + 2 * sqrt(23) - (sqrt(25) + sqrt(92))`**
- We can simplify `sqrt(25)` and `sqrt(92)`:
`sqrt(25) = 5`
`sqrt(92) = sqrt(4 * 23) = 2 * sqrt(23)`
- Therefore, the expression becomes:
`5 + 2 * sqrt(23) - (5 + 2 * sqrt(23)) = 5 + 2 * sqrt(23) - 5 - 2 * sqrt(23) = 0`
- Since 0 is a rational number,
**Conclusion:** `5 + 2 * sqrt(23) - (sqrt(25) + sqrt(92))` is **rational**.

**(viii) `22/7`**
- This is a fraction where both the numerator and denominator are integers.
- Therefore, it is a rational number.
**Conclusion:** `22/7` is **rational**.

**(ix) `pi`**
- The number `pi` is known to be an irrational number.
**Conclusion:** `pi` is **irrational**.

**(x) `3rd root of 37`**
- The cube root of 37 can be expressed as `37^(1/3)`.
- Since 37 is not a perfect cube, `3rd root of 37` is an irrational number.
**Conclusion:** `3rd root of 37` is **irrational**.

### Final Classification:
1. `sqrt(27)` - **Irrational**
2. `sqrt(36)` - **Rational**
3. `sqrt(5) * sqrt(125)` - **Rational**
4. `2 * sqrt(3)` - **Irrational**
5. `(7 * sqrt(7)) / (sqrt(343))` - **Rational**
6. `2 + sqrt(21)` - **Irrational**
7. `5 + 2 * sqrt(23) - (sqrt(25) + sqrt(92))` - **Rational**
8. `22/7` - **Rational**
9. `pi` - **Irrational**
10. `3rd root of 37` - **Irrational**