Case III.

11. $49a^2{b}^2-4$.

12. $25a^4{b}^4-9$.

13. $9a^8{b}^6-16x^{10}$.

14. $144x^2{y}^2-1$.

15. $100x^6{y}^2{z}^4-1$.

16. $1-121a^4{b}^8{c}^{12}$.

17. $25a^2-64x^6{y}^6$.

18. $16x^{16}-25y^{18}$.

Find, by resolving into factors, the value of:

19. $(375{)}^2-(225{)}^2$.

20. $(579{)}^2-(559{)}^2$.

21. $(873{)}^2-(173{)}^2$.

22. $(101{)}^2-(99{)}^2$.

23. $(7244{)}^2-(7242{)}^2$.

24. $(3781{)}^2-(219{)}^2$.

If the squares are compound expressions, the same method may be employed.

1. Resolve into factors $(x+3y{)}^2-16a^2$.

The square root of the first term is $x+3y$.

The square root of the second term is $4a$.

The sum of these roots is $x+3y-4a$.

The difference of these roots is $x+3y-4a$.

Therefore $(x+3y{)}^2-16a^2=(x+3y+4a)(x+3y-4a)$.

2. Resolve into factors $a^2-(3b-5c{)}^2$.

The square roots of the terms are $a$ and $(3b-5c)$.

The sum of these roots is $a+(3b-5c)$, or $a+3b-5c$.

The difference of these roots is $a-(3b-5c)$, or $a-3b+5c$.