If fraction numerator square root of square root of square root of square root of 5 plus 2 to the power of x end root end root end root end root over denominator 2 to the power of x minus 1 end fraction equals 1 over 3 to the power of begin display style bevelled 7 over 8 end style end exponent, then find the value of x.

A) 4

B) 1

C) 2

D)3 

Asked by adalroshan2464 | 14th Nov, 2018, 06:16: PM

Expert Answer:

Given equation can be written as,   begin mathsize 12px style fraction numerator open parentheses 5 plus 2 to the power of x close parentheses to the power of begin display style bevelled 1 over 16 end style end exponent over denominator 2 to the power of x minus 1 end fraction space equals space 1 over 3 to the power of begin display style bevelled 7 over 8 end style end exponent space................... left parenthesis 1 right parenthesis
end style
By cross multiplication, we write,    begin mathsize 12px style open parentheses 5 plus 2 to the power of x close parentheses to the power of bevelled 1 over 16 end exponent space cross times space 3 to the power of bevelled 7 over 8 end exponent space equals space open parentheses space 2 to the power of x space minus space 1 space close parentheses space..................... left parenthesis 2 right parenthesis end style
Raising both sides to the power of 16 on both side, we get, begin mathsize 12px style open parentheses 5 plus 2 to the power of x close parentheses space cross times space 3 to the power of 14 space equals space open parentheses space 2 to the power of x space minus space 1 space close parentheses to the power of 16 space..................... left parenthesis 3 right parenthesis end style
eqn.(3) can be solved only if all the terms are expressed as powers of 3.
Hence ( 2x - 1 ) = 3  or x =2 .  Also ( 5 + 2x ) = 9
 
hence LHS of eqn.(13)  9×314 = 316 =  RHS of eqn.(13)
 
Hence the answer is x = 2

Answered by Thiyagarajan K | 17th Nov, 2018, 04:00: AM

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