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Example 6. If xy 2 and xy 24 , what is the value of xy?

Solution :(xy)^2 (xy)^2  4 xy ( 2 )^2  4 u 24 4  96 100
? xy r 100 r 10

Example 7. If a^4 a^2 b^2 b^4 3 and a^2 abb^2 3 , what is the value of
a^2 b^2?
Solution :a^4 a^2 b^2 b^4 (a^2 )^2  2 a^2 b^2 (b^2 )^2 a^2 b^2
= (a^2 b^2 )^2 (ab)^2
= (a^2 b^2 ab)(a^2 b^2 ab)
= (a^2 abb^2 )(a^2 abb^2 )
? 3 3 (a^2 abb^2 ) [substituting the values]

or, 1
3

2 2 3

a abb

Now adding, a^2 abb^2 3 and a^2 abb^2 1 we get, 2 (a^2 b^2 ) 4

or, 2
2

2 2 4

a b

?a^2 b^2 2

Example 8. Prove that, (ab)^4 (ab)^4 8 ab(a^2 b^2 )

Solution :(ab)^4 (ab)^4 {(ab)^2 }^2 {( ab)^2 }^2

= {(ab)^2 (ab)^2 }{( ab)^2 (ab)^2 }

= 2 (a^2 b^2 )u 4 ab

[(ab)^2 (ab)^2 2 (a^2 b^2 ) and (ab)^2 (ab)^2 4 ab]

= 8 ab(a^2 b^2 )

?(ab)^4 (ab)^4 8 ab(a^2 b^2 )

Example 9. If abc 15 and a^2 b^2 c^2 83 , what is the value of
abbcac?
Soluton :
Here, 2 (abbcac)

(abc)^2 (a^2 b^2 c^2 )

= ( 15 )^2  83

= 225  83
= 142

? 71
2

142

abbcac

Alternative method,

We know ,

(abc)^2 (a^2 b^2 c^2 ) 2 (abbcac)

or,( 15 )^2 83  2 (abbcac)

or, 225  83 2 (abbcac)

or, 2 (abbcac) 142

? 71

2

142

abbcac