WMC 2018 Senior Problem 19

Algebra Level 2

If 5 − x = 6 5^{-x}=6 , then 25 2 x + 1 = ? {25}^{2x+1}= ?

None of these 25 1296 \frac{25}{1296} 216 625 \frac{216}{625} 1296 25 \frac{1296}{25} 625 216 \frac{625}{216}

Tiger Ang by Tiger Ang , 20, Malaysia

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2 solutions

Chew-Seong Cheong
Oct 15, 2018

Given that 5 − x = 6 5^{-x} = 6 , ⟹ 5 x = 1 6 \implies 5^x = \dfrac 16 . Then 2 5 2 x + 1 = 5 4 x + 2 = 25 × 5 4 x = 25 6 4 = 25 1296 25^{2x+1} = 5^{4x+2} = 25\times 5^{4x} = \dfrac {25}{6^4} = \boxed{\dfrac {25}{1296}} .

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X X
Oct 14, 2018

2 5 2 x + 1 = 5 4 x × 25 = ( 5 x ) 4 × 25 = 25 6 4 25^{2x+1}=5^{4x}\times25=(5^x)^4\times25=\dfrac{25}{6^4}

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