Negative indices

Number Theory Level 2

( 3 − 1 + 2 − 2 2 − 1 + 3 − 2 ) − 1 \large \left (\dfrac {3^{-1} + 2^{-2}}{2^{-1} + 3^{-2}} \right )^{-1}

If the above expression can be simplified as a b \dfrac {a}{b} , where a a and b b are coprime positive integers, find a + b a+b .


The answer is 43.

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Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Chung Kevin
Nov 21, 2016

We have to be very careful and evaluate each step one at a time.

( 3 − 1 + 2 − 2 2 − 1 + 3 − 2 ) − 1 = ( 1 3 + 1 4 1 2 + 1 9 ) − 1 = ( 7 12 11 18 ) − 1 = ( 21 22 ) − 1 = 22 21 \begin{array} { l l } \left (\dfrac {3^{-1} + 2^{-2}}{2^{-1} + 3^{-2}} \right )^{-1} & = \left( \frac{ \frac{ 1}{3} + \frac{1}{4} } { \frac{ 1}{2} + \frac{1}{9} } \right)^{-1} \\ & = \left( \frac{ \frac{7}{12} } { \frac{ 11}{18} } \right)^{-1} \\ & = \left( \frac{21}{22} \right)^{-1} \\ & = \frac{22}{21} \\ \end{array}

Hence, the answer is 22 + 21 = 43 22 + 21 = 43 .

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