To solve the given products using suitable identities, we will apply the following algebraic identities:
1. \( (a + b)^2 = a^2 + 2ab + b^2 \)
2. \( (a - b)^2 = a^2 - 2ab + b^2 \)
3. \( (a + b)(a - b) = a^2 - b^2 \)
Now, let's solve each part step by step.
### (i) \( (x + 3)(x + 3) \)
Using the identity \( (a + b)^2 \):
- Here, \( a = x \) and \( b = 3 \).
- Therefore, \( (x + 3)^2 = x^2 + 2(x)(3) + 3^2 \).
- Simplifying gives: \( x^2 + 6x + 9 \).
**Hint:** Recognize that multiplying a binomial by itself is the same as squaring it.
### (ii) \( (2y + 5)(2y + 5) \)
Using the identity \( (a + b)^2 \):
- Here, \( a = 2y \) and \( b = 5 \).
- Therefore, \( (2y + 5)^2 = (2y)^2 + 2(2y)(5) + 5^2 \).
- Simplifying gives: \( 4y^2 + 20y + 25 \).
**Hint:** Identify the coefficients and apply the square of a binomial.
### (iii) \( (2a - 7)(2a - 7) \)
Using the identity \( (a - b)^2 \):
- Here, \( a = 2a \) and \( b = 7 \).
- Therefore, \( (2a - 7)^2 = (2a)^2 - 2(2a)(7) + 7^2 \).
- Simplifying gives: \( 4a^2 - 28a + 49 \).
**Hint:** Remember that the square of a binomial includes a negative middle term.
### (iv) \( (3a - \frac{1}{2})(3a - \frac{1}{2}) \)
Using the identity \( (a - b)^2 \):
- Here, \( a = 3a \) and \( b = \frac{1}{2} \).
- Therefore, \( (3a - \frac{1}{2})^2 = (3a)^2 - 2(3a)(\frac{1}{2}) + \left(\frac{1}{2}\right)^2 \).
- Simplifying gives: \( 9a^2 - 3a + \frac{1}{4} \).
**Hint:** Pay attention to the fractions when squaring.
### (v) \( (1.1m - 0.4)(1.1m + 0.4) \)
Using the identity \( (a - b)(a + b) \):
- Here, \( a = 1.1m \) and \( b = 0.4 \).
- Therefore, \( (1.1m)^2 - (0.4)^2 \).
- Simplifying gives: \( 1.21m^2 - 0.16 \).
**Hint:** This is a difference of squares; remember the formula.
### (vi) \( (a^2 + b^2)(-a^2 + b^2) \)
Using the identity \( (a + b)(a - b) \):
- Here, \( a = b^2 \) and \( b = a^2 \).
- Therefore, \( (b^2)^2 - (a^2)^2 \).
- Simplifying gives: \( b^4 - a^4 \).
**Hint:** Rearranging terms can help identify the correct identity.
### (vii) \( (6x - 7)(6x + 7) \)
Using the identity \( (a - b)(a + b) \):
- Here, \( a = 6x \) and \( b = 7 \).
- Therefore, \( (6x)^2 - (7)^2 \).
- Simplifying gives: \( 36x^2 - 49 \).
**Hint:** Look for the pattern of a difference of squares.
### (viii) \( \left(\frac{x}{2} + \frac{3y}{4}\right)\left(\frac{x}{2} + \frac{3y}{4}\right) \)
Using the identity \( (a + b)^2 \):
- Here, \( a = \frac{x}{2} \) and \( b = \frac{3y}{4} \).
- Therefore, \( \left(\frac{x}{2}\right)^2 + 2\left(\frac{x}{2}\right)\left(\frac{3y}{4}\right) + \left(\frac{3y}{4}\right)^2 \).
- Simplifying gives: \( \frac{x^2}{4} + \frac{3xy}{4} + \frac{9y^2}{16} \).
**Hint:** Be careful with fractions when applying the square of a binomial.
### (ix) \( (7a - 9b)(7a - 9b) \)
Using the identity \( (a - b)^2 \):
- Here, \( a = 7a \) and \( b = 9b \).
- Therefore, \( (7a)^2 - 2(7a)(9b) + (9b)^2 \).
- Simplifying gives: \( 49a^2 - 126ab + 81b^2 \).
**Hint:** Ensure you correctly apply the coefficients in the identity.
