1. ## Proving/manipulation question

Given that (3^(x+2))*(5^(x-1)) = 15^(2x), show that 15^x = 9/5.

I managed to reach: 15^x= [3*sqrt(5)*sqrt(3^x)*sqrt(5^x)]/5, but can't reach 9/5.

2. Are you sure you copied the question correctly? as:

$\displaystyle 15x=\frac{9}{5}$

$\displaystyle \therefore x=\frac{3}{25}$

So,

$\displaystyle \left (3\times\frac{3}{25}+2\right )\left (5\times \frac{3}{25}-1\right )$ must equal $\displaystyle 152\times\frac{3}{25}$

but this gives $\displaystyle -\frac{118}{125}=\frac{456}{25}$, which is false.

3. Sorry, the statement was altered during the posting process. I copied the original statement (with superscript) but during posting it was altered to normal text.

The wrong statement has been corrected.

4. $\displaystyle \displaystyle ({3^{x+2}})(5^{x-1}) = 15^{2x}$ $\displaystyle \displaystyle \Rightarrow (3^{x+2})(5^{x-1}) = 15^{x}\cdot 15^{x}$ $\displaystyle \displaystyle \Rightarrow \frac{3^x\cdot 3^2 \cdot 5^{x}\cdot 5^{-1}}{15^x} = 15^{x}$ $\displaystyle \displaystyle\Rightarrow \frac{9}{5}\left(\frac{3^{x}\cdot 5^{x}}{15^x}\right) = 15^{x}$.

5. How do you know to do it this way rather than square rooting both sides?

6. Originally Posted by fterh
How do you know to do it this way rather than square rooting both sides?
Which step are you referring to? I didn't take any square roots. You understand that $\displaystyle x^{a+b} = x^ax^b$, right?

7. Originally Posted by fterh
Given that (3^(x+2))*(5^(x-1)) = 15^(2x), show that 15^x = 9/5.

I managed to reach: 15^x= [3*sqrt(5)*sqrt(3^x)*sqrt(5^x)]/5, but can't reach 9/5.
$\displaystyle 3^{x + 2}\cdot 5^{x - 1} = 15^{2x}$

$\displaystyle \ln{\left(3^{x + 2}\cdot 5^{x - 1}\right)} = \ln{\left(15^{2x}\right)}$

$\displaystyle \ln{\left(3^{x + 2}\right)} + \ln{\left(5^{x - 1}\right)} = 2x\ln{(15)}$

$\displaystyle (x + 2)\ln{3} + (x - 1)\ln{5} = 2x(\ln{3} + \ln{5})$

$\displaystyle x\ln{3} + 2\ln{3} + x\ln{5} - \ln{5} = 2x\ln{3} + 2x\ln{5}$

$\displaystyle 2\ln{3} - \ln{5} = x\ln{3} + x\ln{5}$

$\displaystyle 2\ln{3} - \ln{5} = x(\ln{3} + \ln{5})$

$\displaystyle \frac{2\ln{3} - \ln{5}}{\ln{3} + \ln{5}} = x$.

8. Originally Posted by fterh
How do you know to do it this way rather than square rooting both sides?
I think I misread your question earlier. Apologies. The answer is because taking the square root of both sides is not going to take you anywhere.
There is no method of identifying which technique to use what time; it's just insights that one gains from exposure and experience by doing lots of this.

9. Thanks a lot, gonna mark this as Solved. I guess the key is to practice and gain extensive exposure, huh.