1. ## Two problems about indices and exponential equations

Could someone please help me solve them? These were on my test, but I didn't manage to do them.

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

2) if $2^{x} =15$ and $15^{y}=32$
How much is $xy$?

Thank you!

2. Originally Posted by Coach
2) if $2^{x} =15$ and $15^{y}=32$
How much is $xy$?
$x = log_{2} 15$

$y = log_{15} 32$

$x = \frac{1}{ log_{15} 2 }$

$xy = \frac{ log_{15} 32 }{ log_{15} 2 }$

$xy = 5$

3. Thank you!

By the way, could one use the zero product property for the first exponential equation?

4. Originally Posted by Coach
Thank you!

By the way, could one use the zero product property for the first exponential equation?
I'm not quite sure what you mean by that, give me an example of how it is used?

5. Originally Posted by janvdl
I'm not quite sure what you mean by that, give me an example of how it is used?
i believe he is talking about setting the power equal to zero (since anything raised to zero is 1). i think that's the way to go

6. Originally Posted by Coach

1) $(x^{2}-5x+5)^{(x^{2}-9)}=1$
So $x = +3$ or $x = -3$ ?

But that $= 1$ wasn't there earlier...

7. Originally Posted by janvdl
So $x = +3$ or $x = -3$ ?
yes

But that $= 1$ wasn't there earlier...
yes, that happens from time to time. the instructions were to "solve" so there should have been = something, so if it wasn't there, we would know that something was off with the question

8. I tried expanding this:

$(x^{2}-5x+5)^{(x^{2}-9)}$

Because I thought it needed to be expanded...

By the way, is it anyhow possible to expand something like this?

9. Originally Posted by janvdl
I tried expanding this:

$(x^{2}-5x+5)^{(x^{2}-9)}$

Because I thought it needed to be expanded...

By the way, is it anyhow possible to expand something like this?
i very much doubt it. i would say no, but i've been seeing some things done in math lately that i would have said was impossible, so it's a "likely" no

curious: how did you try expanding it?

10. Originally Posted by Jhevon
i very much doubt it. i would say no, but i've been seeing some things done in math lately that i would have said was impossible, so it's a "likely" no

curious: how did you try expanding it?
I tried setting values into $x$ and expanding it, and trying to note patterns. Like with series and sequences, if you understand my method.

Maybe CaptainBlack or TPH can do it...

11. Yes, I forgot the one at first, I'm sorry.

Yes I also thought about the zero product property,as $1^{0}=1$

But I guess one could raise each term separately into the power.

12. Originally Posted by Coach
But I guess one could raise each term separately into the power.
You mean $(a + b)^{a + b} = a^{a + b} + b^{a + b}$ ?

That's wrong.

EDIT: Remember $(a + b)^{2} = a^{2} + 2ab + b^{2}$ ?

13. Originally Posted by Coach
Well, I was just guessing.

But good to know that. Thank you!
Dont worry, i tried doing that, before realising it was wrong

14. Is there any way one could multiply or raise to a power both sides to eliminate the power?