Evaluate using formula :
`(i) (188)^(2) " " (ii) (9.4)^(2) " " (iii) (10.9)^(2)`

Answer

To evaluate the squares of the given numbers using the formulas, we will apply the identities for the square of a binomial. The two identities we will use are:

1. \( (A + B)^2 = A^2 + 2AB + B^2 \)
2. \( (A - B)^2 = A^2 - 2AB + B^2 \)

Now, let's solve each part step by step.

### (i) Evaluate \( (188)^2 \)

1. **Identify A and B**: We can express 188 as \( 200 - 12 \).
- Here, \( A = 200 \) and \( B = 12 \).

2. **Apply the formula**: Use the identity for \( (A - B)^2 \):
\[ (A - B)^2 = A^2 - 2AB + B^2 \]
Thus,
\[ (188)^2 = (200 - 12)^2 = 200^2 - 2 \cdot 200 \cdot 12 + 12^2 \]

3. **Calculate each term**:
- \( 200^2 = 40000 \)
- \( 2 \cdot 200 \cdot 12 = 4800 \)
- \( 12^2 = 144 \)

4. **Combine the results**:
\[ (188)^2 = 40000 - 4800 + 144 = 35200 + 144 = 35344 \]

### (ii) Evaluate \( (9.4)^2 \)

1. **Identify A and B**: We can express 9.4 as \( 10 - 0.6 \).
- Here, \( A = 10 \) and \( B = 0.6 \).

2. **Apply the formula**: Use the identity for \( (A - B)^2 \):
\[ (9.4)^2 = (10 - 0.6)^2 = 10^2 - 2 \cdot 10 \cdot 0.6 + 0.6^2 \]

3. **Calculate each term**:
- \( 10^2 = 100 \)
- \( 2 \cdot 10 \cdot 0.6 = 12 \)
- \( 0.6^2 = 0.36 \)

4. **Combine the results**:
\[ (9.4)^2 = 100 - 12 + 0.36 = 88 + 0.36 = 88.36 \]

### (iii) Evaluate \( (10.9)^2 \)

1. **Identify A and B**: We can express 10.9 as \( 10 + 0.9 \).
- Here, \( A = 10 \) and \( B = 0.9 \).

2. **Apply the formula**: Use the identity for \( (A + B)^2 \):
\[ (10.9)^2 = (10 + 0.9)^2 = 10^2 + 2 \cdot 10 \cdot 0.9 + 0.9^2 \]

3. **Calculate each term**:
- \( 10^2 = 100 \)
- \( 2 \cdot 10 \cdot 0.9 = 18 \)
- \( 0.9^2 = 0.81 \)

4. **Combine the results**:
\[ (10.9)^2 = 100 + 18 + 0.81 = 118 + 0.81 = 118.81 \]

### Final Results:
- \( (188)^2 = 35344 \)
- \( (9.4)^2 = 88.36 \)
- \( (10.9)^2 = 118.81 \)