i)  Let x= 0.23 (23 repeating)

   x=0.2323.....--------------(i)

  (Multiplying by 100 on both sides)

  100x= 23.2323.....--------(ii)

  subtracting eq-(i) from eq-(ii)

  99x=23

  So x= 23/99

ii)  { x^[a(b - c)] / x^[b(a - c)] } / (x^b / x^a)^c = 1

    [ x^(ab - ac) / x^(ab - bc) ] / [x^(bc) / x^(ac)] = 1

    [ x^(ab - ac) / x^(ab - bc) ] * [x^(ac) / x^(bc)] = 1

    x^[(ab - ac) - (ab - bc)] * [x^(ac - bc)] = 1

    x^(ab - ac - ab + bc) * x^(ac - bc) = 1

    x^(-ac + bc) * x^(ac - bc) = 1

    x^[(-ac + bc) + (ac - bc)] = 1

    x^(-ac + bc + ac - bc) = 1

    x^(0) = 1  (since the power is zero, x becomes 1 )

    PROVED!!!

HOPE IT WILL HELP YOU FRIEND...

ALL THE BEST FOR YOUR TEST, SCORE GOOD :-)

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Step-by-step explanation:

i hope it is help full for u