#3**+1 **

Why are you having problems with this simple question? If your calculator display isn't large enough, then use the calculator on this site!

81^n=9^27 * 27^9?

1 - With a calculator, calculate the right-hand side first: 9^27 * 27^9 =443426488 2430377699 4824963061 9149892803

2 - Take the log (base 10) of the above number: log(443426488 2430377699 4824963061 9149892803)

3 - Log = 38.6468216322 (you only need the log to 10 decimal places).

4 - Take the log of 81 to 10 decimal places: log(81) = 1.9084850189

5 - Divide the log in (3) above by the log in (4) above: 38.6468216322 / 1.9084850189=20.25

**6 - Therefore: n = 20.25. And that is it!**

Guest Dec 30, 2022

#4**+1 **

I'm thinking to solve it a different way...

We have:

\(81^n=9^{27} \cdot 27^9\)

We know 81 is 3^4

\((3^4)^n = 9^{27} \cdot 27^9\)

So the left side simplifies to

\(3^{4n} = 9^{27} \cdot 27^9\)

We know 9 is 3^2 and 27 is 3^3 so we use that information into our equation:

\(3^{4n} = (3^2)^{27} \cdot (3^3)^9\)

So the right side simplifies to:

\(3^{4n} = (3^{54}) \cdot (3^{27})\)

Then, we know that if we have the same base when multiplying, we add the exponents so:

\(3^{4n} = (3^{81})\)

So:

\(4n = 81 \)

Divide by 4 to both sides:

\(n = 81/4 = 20.25\)

This is (I believe) the **standard** and **faster** way to solve it in a real competition.

TooEasy Dec 30, 2022