( m n ) 3 (\frac{m}{n})^3

Algebra Level 2

1 8 12 1 2 18 = ( m n ) 3 , \frac{18^{12}}{12^{18}}=\Big(\frac{m}{n}\Big)^3,

where m m and n n are coprime positive integers. What is m + n m+n ?

245 265 256 260 254

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Ram Mohith
Oct 7, 2018

1 8 12 1 2 18 = ( 1 8 4 1 2 6 ) 3 = ( 3 4 × 3 4 × 2 4 3 6 × 4 6 ) 3 = ( 3 2 4 4 ) 3 = ( 9 256 ) 3 \dfrac{18^{12}}{12^{18}} = \left ( \dfrac{18^4}{12^6} \right )^3 = \left (\dfrac{3^4 \times 3^4 \times 2^4 }{3^6 \times 4^6} \right)^3 = \left ( \dfrac{3^2}{4^4} \right )^3 = \left ( \dfrac{9}{256} \right )^3

∴ m + n = 9 + 256 = 265 \therefore m + n = 9 + 256 = 265

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you @Ram Mohith, l also see the solution has been edited. This one has more explanation which makes it better.

Hana Wehbi - 2 years, 8 months ago

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