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If \(\left(\sqrt[4]{11}\right)^{3x-3}=\frac{1}{5}\), what is the value of \(\left(\sqrt[4]{11}\right)^{6x+2}\)? Express your answer as a fraction.

off-topic
michaelcai  Sep 16, 2017
edited by michaelcai  Sep 16, 2017
 #1
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I hope this helps ( i took a snapshot from word, cause LaTeX gave me cancer cheeky).

DrDros  Sep 16, 2017
 #2
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Thanks, Dr Dros!!!

 

Here's another method without resorting to logs

 

[(11)^ (1/4)] ^(3x -3)  = 1/5     implies that

 

[  11 ^(x - 1) ] ^ (3/4)   = 1/5       take both sides to the 4/3 power

 

[11 ^(x - 1) ]   =  (1/5)^(4/3)

 

[11^x ] / 11  =   (1/5)^(4/3)

 

11^x  =  11* (1/5)^(4/3)

 

So.....  11^(6x)  =  (11^x)^6  =  [ 11 * (1/5)^(4/3) ]^6  =  [ 11^6] * [ 1/5]^8

 

So..... [11^(1/4)]^(6x + 2)  =   [ 11^(6x + 2) ] ^(1/4)  =  [ 11 ^(6x) * 11^2] ^ (1/4)  =

 

 [ 11^6 * (1/5)^8 ] ^(1/4) *  11^(1/2)   =

 

[ 11^(6/4)] * [(1/5)^8]^(1/4) * 11^(1/2)  =

 

[11 ^ (3/2) * [ (1/5) ^2] * 11^(1/2) =

 

[ 11 ^2]  *  (1 / 25)  =

 

121  / 25

 

 

cool cool cool

CPhill  Sep 16, 2017

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