(i) 64 a 2 + 9 b 2 + 48 a b (ii) 49 x 2 + 4 y 2 + 28 x y (iii) 25 x 2 + 110 x + 121 (iv) a 2 4 + 4 a 2 + 2 (v) 9 x 2 16 + 4 y 2 81 + 1 3 x y (vi) 81 x 2 − 180 x + 100 (vii) x 4 y 2 + y 2 z 4 − 2 x 2 y 2 z 2 (viii) x 2 y 2 + y 2 x 2 − 2 (ix) 9 m 2 + 16 25 n 2 − 24 5 m n

Answer

To solve the given algebraic expressions, we will use the formulas for the square of a binomial:

1. For \( (a + b)^2 = a^2 + 2ab + b^2 \)
2. For \( (a - b)^2 = a^2 - 2ab + b^2 \)

Now, let's solve each expression step by step:

### (i) \( 64a^2 + 9b^2 + 48ab \)

1. Identify \( a \) and \( b \):
- \( a = 8a \)
- \( b = 3b \)

2. Apply the formula:
\[ (8a + 3b)^2 = (8a)^2 + 2(8a)(3b) + (3b)^2 \]

3. Calculate:
- \( (8a)^2 = 64a^2 \)
- \( 2(8a)(3b) = 48ab \)
- \( (3b)^2 = 9b^2 \)

4. Combine:
\[ (8a + 3b)^2 = 64a^2 + 48ab + 9b^2 \]

### (ii) \( 49x^2 + 4y^2 + 28xy \)

1. Identify \( a \) and \( b \):
- \( a = 7x \)
- \( b = 2y \)

2. Apply the formula:
\[ (7x + 2y)^2 = (7x)^2 + 2(7x)(2y) + (2y)^2 \]

3. Calculate:
- \( (7x)^2 = 49x^2 \)
- \( 2(7x)(2y) = 28xy \)
- \( (2y)^2 = 4y^2 \)

4. Combine:
\[ (7x + 2y)^2 = 49x^2 + 28xy + 4y^2 \]

### (iii) \( 25x^2 + 110x + 121 \)

1. Identify \( a \) and \( b \):
- \( a = 5x \)
- \( b = 11 \)

2. Apply the formula:
\[ (5x + 11)^2 = (5x)^2 + 2(5x)(11) + (11)^2 \]

3. Calculate:
- \( (5x)^2 = 25x^2 \)
- \( 2(5x)(11) = 110x \)
- \( (11)^2 = 121 \)

4. Combine:
\[ (5x + 11)^2 = 25x^2 + 110x + 121 \]

### (iv) \( \frac{a}{2} + 2 \)

1. Identify \( a \) and \( b \):
- \( a = \frac{a}{2} \)
- \( b = 2 \)

2. Apply the formula:
\[ \left(\frac{a}{2} + 2\right)^2 = \left(\frac{a}{2}\right)^2 + 2\left(\frac{a}{2}\right)(2) + (2)^2 \]

3. Calculate:
- \( \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \)
- \( 2\left(\frac{a}{2}\right)(2) = 2a \)
- \( (2)^2 = 4 \)

4. Combine:
\[ \left(\frac{a}{2} + 2\right)^2 = \frac{a^2}{4} + 2a + 4 \]

### (v) \( 9x^2 + 16y^2 + 81 \)

1. Identify \( a \) and \( b \):
- \( a = \frac{3x}{4} \)
- \( b = \frac{2y}{9} \)

2. Apply the formula:
\[ \left(\frac{3x}{4} + \frac{2y}{9}\right)^2 = \left(\frac{3x}{4}\right)^2 + 2\left(\frac{3x}{4}\right)\left(\frac{2y}{9}\right) + \left(\frac{2y}{9}\right)^2 \]

3. Calculate:
- \( \left(\frac{3x}{4}\right)^2 = \frac{9x^2}{16} \)
- \( 2\left(\frac{3x}{4}\right)\left(\frac{2y}{9}\right) = \frac{12xy}{18} = \frac{2xy}{3} \)
- \( \left(\frac{2y}{9}\right)^2 = \frac{4y^2}{81} \)

4. Combine:
\[ \left(\frac{3x}{4} + \frac{2y}{9}\right)^2 = \frac{9x^2}{16} + \frac{2xy}{3} + \frac{4y^2}{81} \]

### (vi) \( 81x^2 - 180x + 100 \)

1. Identify \( a \) and \( b \):
- \( a = 9x \)
- \( b = 10 \)

2. Apply the formula:
\[ (9x - 10)^2 = (9x)^2 - 2(9x)(10) + (10)^2 \]

3. Calculate:
- \( (9x)^2 = 81x^2 \)
- \( -2(9x)(10) = -180x \)
- \( (10)^2 = 100 \)

4. Combine:
\[ (9x - 10)^2 = 81x^2 - 180x + 100 \]

### (vii) \( x^4y^2 + y^2z^4 - 2x^2y^2z^2 \)

1. Identify \( a \) and \( b \):
- \( a = x^2y \)
- \( b = yz^2 \)

2. Apply the formula:
\[ (x^2y - yz^2)^2 = (x^2y)^2 - 2(x^2y)(yz^2) + (yz^2)^2 \]

3. Calculate:
- \( (x^2y)^2 = x^4y^2 \)
- \( -2(x^2y)(yz^2) = -2x^2y^2z^2 \)
- \( (yz^2)^2 = y^2z^4 \)

4. Combine:
\[ (x^2y - yz^2)^2 = x^4y^2 - 2x^2y^2z^2 + y^2z^4 \]

### (viii) \( x^2y^2 + y^2x^2 - 2 \)

1. Identify \( a \) and \( b \):
- \( a = xy \)
- \( b = xy \)

2. Apply the formula:
\[ (xy - xy)^2 = (xy)^2 - 2(xy)(xy) + (xy)^2 \]

3. Calculate:
- \( (xy)^2 = x^2y^2 \)
- \( -2(xy)(xy) = -2x^2y^2 \)

4. Combine:
\[ (xy - xy)^2 = 0 \]

### (ix) \( 9m^2 + \frac{16}{25}n^2 - \frac{24}{5}mn \)

1. Identify \( a \) and \( b \):
- \( a = 3m \)
- \( b = \frac{4}{5}n \)

2. Apply the formula:
\[ (3m - \frac{4}{5}n)^2 = (3m)^2 - 2(3m)(\frac{4}{5}n) + (\frac{4}{5}n)^2 \]

3. Calculate:
- \( (3m)^2 = 9m^2 \)
- \( -2(3m)(\frac{4}{5}n) = -\frac{24}{5}mn \)
- \( (\frac{4}{5}n)^2 = \frac{16}{25}n^2 \)

4. Combine:
\[ (3m - \frac{4}{5}n)^2 = 9m^2 - \frac{24}{5}mn + \frac{16}{25}n^2 \]