Use direct method to evaluate :
`(i) (x+y)(x-y) " " (ii) (4+x)(4-x) " " (iii) (2x-1)(2x+1) " " (iv) (xy+a)(xy-a)`
`(v) ((x)/(2)-(y)/(2))((x)/(2)+(y)/(3)) " " (vi) (x+y)(x-y)(x^(2)+y^(2))`.

Answer

To solve the given problems using the direct method, we will apply the identity \( (a + b)(a - b) = a^2 - b^2 \) for each part.

### Solution:

**(i)** Evaluate \( (x+y)(x-y) \):
Using the identity:
\[ (x+y)(x-y) = x^2 - y^2 \]
**Answer:** \( x^2 - y^2 \)

**(ii)** Evaluate \( (4+x)(4-x) \):
Using the identity:
\[ (4+x)(4-x) = 4^2 - x^2 = 16 - x^2 \]
**Answer:** \( 16 - x^2 \)

**(iii)** Evaluate \( (2x-1)(2x+1) \):
Using the identity:
\[ (2x-1)(2x+1) = (2x)^2 - 1^2 = 4x^2 - 1 \]
**Answer:** \( 4x^2 - 1 \)

**(iv)** Evaluate \( (xy+a)(xy-a) \):
Using the identity:
\[ (xy+a)(xy-a) = (xy)^2 - a^2 = x^2y^2 - a^2 \]
**Answer:** \( x^2y^2 - a^2 \)

**(v)** Evaluate \( \left(\frac{x}{2}-\frac{y}{2}\right)\left(\frac{x}{2}+\frac{y}{3}\right) \):
Using the identity:
\[ \left(\frac{x}{2}-\frac{y}{2}\right)\left(\frac{x}{2}+\frac{y}{3}\right) = \left(\frac{x}{2}\right)^2 - \left(\frac{y}{3}\right)^2 = \frac{x^2}{4} - \frac{y^2}{9} \]
**Answer:** \( \frac{x^2}{4} - \frac{y^2}{9} \)

**(vi)** Evaluate \( (x+y)(x-y)(x^2+y^2) \):
First, apply the identity to \( (x+y)(x-y) \):
\[ (x+y)(x-y) = x^2 - y^2 \]
Now, multiply by \( (x^2 + y^2) \):
\[ (x^2 - y^2)(x^2 + y^2) = (x^2)^2 - (y^2)^2 = x^4 - y^4 \]
**Answer:** \( x^4 - y^4 \)

### Summary of Answers:
1. \( x^2 - y^2 \)
2. \( 16 - x^2 \)
3. \( 4x^2 - 1 \)
4. \( x^2y^2 - a^2 \)
5. \( \frac{x^2}{4} - \frac{y^2}{9} \)
6. \( x^4 - y^4 \)