## Poll

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**3 members have voted**

March 15th, 2020 at 4:53:16 PM
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The question is simple:

9

Find x.

9

^{x}+ 12^{x}= 16^{x}Find x.

It's not whether you win or lose; it's whether or not you had a good bet.

March 15th, 2020 at 5:32:00 PM
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-infinity (actually just approaches equality). Otherwise, there's a solution somewhere between 1 and 2.

I heart Crystal Math.

March 15th, 2020 at 5:38:01 PM
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I’ve never been able to wrap my head around non-whole exponents.

Non-whole roots give me even more headaches.

On the plus side, math problems like this will still be a PITA after the apocalypse. 🤪

Non-whole roots give me even more headaches.

On the plus side, math problems like this will still be a PITA after the apocalypse. 🤪

Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁

March 15th, 2020 at 6:07:51 PM
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I forgot to say the usual rules apply:

Also, I am indeed looking for a real number for credit.

- Please don't just plop a URL to a solution elsewhere until a winner here has been declared.
- All those who have won a beer previously are asked to not post answers or solutions for 24 after this posting. Past winners who must chime in early, may PM me.
- Beer to the first satisfactory answer and solution, subject to rule 2.
- Please put answers and solutions in spoiler tags.

Also, I am indeed looking for a real number for credit.

It's not whether you win or lose; it's whether or not you had a good bet.

March 15th, 2020 at 6:54:09 PM
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As opposed to an imaginary or complex number?Quote:Wizard

Also, I am indeed looking for a real number for credit.

The race is not always to the swift, nor the battle to the strong; but that is the way to bet.

March 15th, 2020 at 7:59:17 PM
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Quote:unJonAs opposed to an imaginary or complex number?

Yep, and infinity.

It's not whether you win or lose; it's whether or not you had a good bet.

March 15th, 2020 at 8:17:16 PM
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9

^{x}+12

^{x}=16

^{x}

Divide everything by 9

^{x}

1+(12/9)

^{x}=(16/9)

^{x}

simplify

1+(4/3)

^{x}=(16/9)

^{x}

recognize that (4/3)

^{x}= (16/9)

^{(x/2)}

1+(16/9)

^{(x/2)}=(16/9)

^{x}

let n = (16/9)

^{(x/2)}

substitute n into above equation

1 + n = n

^{2}

n

^{2}- n - 1 = 0

n = (1+-5

^{(1/2)})/2

(16/9)

^{(x/2)}= (1+-5

^{(1/2)})/2

ln[(16/9)

^{(x/2)}] = ln[(1+-5

^{(1/2)})/2]

(x/2)ln(16/9)=ln[(1+-5

^{(1/2)})/2]

x = 2ln[(1+-5

^{(1/2)})/2]/ln(16/9)

It is only valid for the positive root.

x = 2ln[(1+5

^{(1/2)})/2]/ln(16/9) ~=1.672721

Not that it makes any difference here, but (1+5

^{(1/2)})/2 is also the golden ratio.

I heart Crystal Math.

March 15th, 2020 at 8:58:54 PM
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Quote:CrystalMath

9^{x}+12^{x}=16^{x}

Divide everything by 9^{x}

1+(12/9)^{x}=(16/9)^{x}

simplify

1+(4/3)^{x}=(16/9)^{x}

recognize that (4/3)^{x}= (16/9)^{(x/2)}

1+(16/9)^{(x/2)}=(16/9)^{x}

let n = (16/9)^{(x/2)}

substitute n into above equation

1 + n = n^{2}

n^{2}- n - 1 = 0

n = (1+-5^{(1/2)})/2

(16/9)^{(x/2)}= (1+-5^{(1/2)})/2

ln[(16/9)^{(x/2)}] = ln[(1+-5^{(1/2)})/2]

(x/2)ln(16/9)=ln[(1+-5^{(1/2)})/2]

x = 2ln[(1+-5^{(1/2)})/2]/ln(16/9)

It is only valid for the positive root.

x = 2ln[(1+5^{(1/2)})/2]/ln(16/9) ~=1.672721

Not that it makes any difference here, but (1+5^{(1/2)})/2 is also the golden ratio.

I agree. However, weren't you previously a member of the beer club?

It's not whether you win or lose; it's whether or not you had a good bet.

March 15th, 2020 at 9:10:01 PM
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Quote:WizardQuote:CrystalMath

9^{x}+12^{x}=16^{x}

Divide everything by 9^{x}

1+(12/9)^{x}=(16/9)^{x}

simplify

1+(4/3)^{x}=(16/9)^{x}

recognize that (4/3)^{x}= (16/9)^{(x/2)}

1+(16/9)^{(x/2)}=(16/9)^{x}

let n = (16/9)^{(x/2)}

substitute n into above equation

1 + n = n^{2}

n^{2}- n - 1 = 0

n = (1+-5^{(1/2)})/2

(16/9)^{(x/2)}= (1+-5^{(1/2)})/2

ln[(16/9)^{(x/2)}] = ln[(1+-5^{(1/2)})/2]

(x/2)ln(16/9)=ln[(1+-5^{(1/2)})/2]

x = 2ln[(1+-5^{(1/2)})/2]/ln(16/9)

It is only valid for the positive root.

x = 2ln[(1+5^{(1/2)})/2]/ln(16/9) ~=1.672721

Not that it makes any difference here, but (1+5^{(1/2)})/2 is also the golden ratio.I agree. However, weren't you previously a member of the beer club?

I’m almost certain I haven’t participated since these were a contest. Thanks for doing these, since it’s the math that attracted me to this site in the first place.

I heart Crystal Math.

March 16th, 2020 at 12:44:57 PM
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Quote:CrystalMathThanks for doing these, since it’s the math that attracted me to this site in the first place.

You're welcome!

It's not whether you win or lose; it's whether or not you had a good bet.