Find the quotient when (a-1 - 1) is divided by (a - 1).

Given equation is
a^-1 - 1
Apply the law of Fractional Exponents and Laws of Exponents
x^-y = 1/x^y
⇒ a^-1 - 1 = 1/a - 1
⇒ a^-1 - 1 = (1 - a)/a
Now divide by a - 1 in above equation,
⇒ ( a^-1 - 1 ) ÷ (a - 1) = (1 - a)/a ÷ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = (1 - a)/a x 1/ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = -1x (a - 1)/a x 1/ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = -1/a
∴ Required quotient = -1/a.