Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
(3−√a9−a+13−√a−6a2+162729−a3)−1+a(a+9)54⋅
(3−√a9−a+13−√a−6(a2+162)729−a3)−1+a(a+9)54⟶(1)
⇒3−√a9−a=3−√a(3−√a)(3+√a)=13+√a
Now, 13−√a+13−√a=3−√a+3+√a(3+√a)(3−√a)=69−a
⇒6(a2+162)729−a3=6a2+972(9−a)(a2+9a+81)
⇒(3−√a9−a+13−√a−6(a2+162)729−a3)−1=(69−a−(6a2+972)(9−a)(a2+9a+81))−1
⇒(6a2+54a+486−6a2−972(9−a)(a2+9a+81))−1=(54a−486(9−a)(a2+9a+81))−1
⇒(54(a−9)(9−a)(a2+9a+81))−1
⇒(−54a2+9a+81)−1
⇒−(a2+9a+81)54=A
⇒A+B=−(a2+9a)−8154+a(a+9)54
=−8154
=−32.
Hence, solved.