We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
87
4
avatar

Let $$x={4\over{(\sqrt5+1)(\root 4\of5+1)(\root 8\of5+1)(\root {16}\of5+1)}}.$$Find $(x+1)^{48}$.

 May 3, 2019
 #1
avatar+116 
+1

\(x={4\over{(\sqrt5+1)(\root 4\of5+1)(\root 8\of5+1)(\root {16}\of5+1)}}. Find (x+1)^{48}\)

\((x+1)^{48} = 125\) 

 May 3, 2019
 #2
avatar+102466 
+1

Please do not give answers only Anthrax.  

They can get that from an answer page.

I suspect you got this answer from Wolfram|Alpha or some similar site.  

 

 

Ideal we want answers to teach but not to facilitate cheating.

So part answers, or solid hints, without the final answer is usually the best way to go.

Encourage the asker to interact with you.  If they want more imput thay should ask for it.   laugh

Melody  May 3, 2019
edited by Melody  May 3, 2019
edited by Melody  May 3, 2019
 #3
avatar
+2

Solve for x:
4/((sqrt(5) + 1) (5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = x

Multiply numerator and denominator of 4/((sqrt(5) + 1) (5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) by sqrt(5) - 1:
(4 (sqrt(5) - 1))/((sqrt(5) + 1) (5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1) (sqrt(5) - 1)) = x

 

(sqrt(5) + 1) (sqrt(5) - 1) = -1 + 1 sqrt(5) - sqrt(5) + sqrt(5) sqrt(5) = -1 + sqrt(5) - sqrt(5) + 5 = 4:
(4 (sqrt(5) - 1))/(4 (5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = x

 

(4 (sqrt(5) - 1))/(4 (5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = 4/4×(sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = (sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)):


(sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = x

(sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = x is equivalent to x = (sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)):


 x = (sqrt(5) - 1)/((5^(1/4) + 1) (5^(1/8) + 1) (5^(1/16) + 1)) = 0.105823017030235.

[0.105823017030235 +1]^48 =~125

 May 3, 2019
 #4
avatar+102466 
+3

Thanks guest for giving me a hint :)

 

 

\(x={4\over{(\sqrt5+1)(\root 4\of5+1)(\root 8\of5+1)(\root {16}\of5+1)}}\\ x={4\over{(\sqrt5+1)(\root 4\of5+1)(\root 8\of5+1)(\root {16}\of5+1)}}\times \frac{{(\sqrt5-1)(\root 4\of5-1)(\root 8\of5-1)(\root {16}\of5-1)}}{{(\sqrt5-1)(\root 4\of5-1)(\root 8\of5-1)(\root {16}\of5-1)}}\\ x= \frac{4{(\sqrt5-1)(\root 4\of5-1)(\root 8\of5-1)(\root {16}\of5-1)}}{{(5-1)(\root \of5-1)(\root 4\of5-1)(\root {8}\of5-1)}}\\ x= \frac{{(\sqrt5-1)(\root 4\of5-1)(\root 8\of5-1)(\root {16}\of5-1)}}{{(\root \of5-1)(\root 4\of5-1)(\root {8}\of5-1)}}\times \frac{{(\root \of5+1)(\root 4\of5+1)(\root {8}\of5+1)}}{(\root \of5+1)(\root 4\of5+1)(\root {8}\of5+1)}\\ x= \frac{{(5-1)(\root 2\of5-1)(\root 4\of5-1)(\root {16}\of5-1)}}{{(5-1)(\root 2\of5-1)(\root {4}\of5-1)}}\\ x= \frac{{(\root {16}\of5-1)}}{{1}}\\ x=\root {16}\of5-1\\ x+1=\root {16}\of5\\ (x+1)^{48}=(5^{\frac{1}{16}})^{48}\\ (x+1)^{48}=5^3\\ (x+1)^{48}=125\\ \)

.
 May 4, 2019

14 Online Users