The sum of \(\displaystyle{8}^{{{x}}}\) and its reciprocal

fernandoval3sbr

fernandoval3sbr

Answered question

2022-04-09

The sum of 8x and its reciprocal Is 18 by substituting the value of x. My question is, is there a way to solve this within 1 minute? Like a clever shortcut or you just need to brute force this to find the answer?

Answer & Explanation

Varikoseocaz

Varikoseocaz

Beginner2022-04-10Added 9 answers

Step 1
You are given that
a+1a=3
With a=2x and you want to compute
a3+1a3
Just notice that
a3+1a3
=(a+1a)33(a+1a)
=3333=18
Jameson Jensen

Jameson Jensen

Beginner2022-04-11Added 16 answers

Step 1
Suppose that 8x+8x=18. Then, multiplying both sides by 8x, we get 82x+1=18×8x. Moving the product from the right hand side to the left hand side, we get 82x18×8x+1=0. Using the quadratic formula (after substituting u=8x to get u218u+1=0) then gives
8x=u=18±32442=18±3202=18±6452=18±852=9±45.
Finally, both of the solutions for u are positive, so the two solutions for x are log8(9+45) and log8(945).

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