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I was doing this problem and I simplified everything to this:

$$243 = 3^{\frac{m}{10}}$$

However, after that part I got stuck. Because I tried doing this then:

$$\sqrt[10]{243} = 3^m$$

But then I couldn't do anything else so I got stuck.

However on the explanation they give us the hint of using:

$$3 = 243^{\frac{1}{5}}$$

And then I know I could be able to replace the 3 in the original equation and go from there. However I do not understand how 3 is equal to 243 to the power of 1/5, how do you get to that conclusion?

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    $\begingroup$ note that $$3^5=243$$ $\endgroup$ – Dr. Sonnhard Graubner Feb 18 '17 at 20:01
  • $\begingroup$ $81\cdot 3=243$ $\endgroup$ – kingW3 Feb 18 '17 at 20:01
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$$243 = 3^{\frac{m}{10}}$$

$$3^5 = 3^{\frac{m}{10}}$$

$$5 = \frac{m}{10}$$

$$m = 50$$

Also, you made an error in the second step where you supposedly raised both sides by $10$ but instead raised 243 by $\frac {1}{10}$, getting $\sqrt[10]{243}$ instead of $243^{10}$.

$243^{10}=3^m$

$(3^5)^{10}=3^m$

$m= 50$

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Using the fact that given $3 = (243)^{\large \frac 15}$, then $3^5 = \big((243)^{\large \frac 15}\big)^5\implies 243 = 3^5$, we have $$243 = (3^5) = 3^{\frac{m}{10}} \iff ({3^5})^{10} = 3^m$$

$$\iff 3^{50} = 3^m \iff m=50$$

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$$243 = 3^{\frac{m}{10}} \iff 243^{10} = 3^m \iff (3^5)^ {10}=3^m. $$ Can you take it from here?

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You have that $243=3^{m/10}=\left((243)^{1/5}\right)^{m/10}=243^{m/50}$. Therefore, the powers must be equal, giving $1=\frac{m}{50}$, so $m=50$.

Are the steps of this solution manipulations you are comfortable with?

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