Simplifying exponential functions....[i think] :-)

[Guest]

My homework problem says to simplify

I only know how to do the first step, which is to change the square root into an exponent to the 1/2 power.

When I simplify this problem, can I make the numbers 1/9 or is that not simplifying it correctly? How would I solve this problem?

Thank you.

 

[soroban]

Hello, catalinamemday!

What an ugly problem . . .

\(\displaystyle \L\sqrt{\frac{3^{\sqrt{3}+1}}{27}}\)

Simplify the "inside" first: \(\displaystyle \L\:\frac{3^{\sqrt{3}+1}}{3^3} \;=\;3^{\sqrt{3}-2}\)

Take the square root: \(\displaystyle \L\:\sqrt{3^{\sqrt{3}-2}} \;=\;3^{\frac{1}{2}(\sqrt{3}-2)} \;=\;3^{(\frac{1}{2}\sqrt{3}-1)}\)


It's hard to tell when we should stop . . .

We have: \(\displaystyle \L\:\left(3^{\frac{1}{2}\sqrt{3}}\right)\cdot\left(3^{-1}\right) \;=\;\left[\left(3^{\frac{1}{2}}\right)^{\sqrt{3}}\right]\cdot\left(\frac{1}{3}\right) \;=\;\frac{\sqrt{3}^{\sqrt{3}}}{3}\)

 

[Denis]

When I simplify this problem, can I make the numbers 1/9 ...

No.

Hint: 27 = 3^3 ; 1 / 3^3 = 3^(-3)

 

[Guest]

Thanks for the help! I got this one right! Scored a B on the assignment! :-D