Question

How do I solve this problem?

Answer 1

\[\sqrt[56]{9^{70}}\]

Answer 2

I would assume I do 9^70, but i'll get some big number.

Answer 3

The answer to this is \[9\sqrt{3}\]
I just don't know how to get there.

Answer 4

Hey, the following radical can be expressed as\[\Large 9^{\frac{ 70 }{ 56}}=9^{\frac{ 5 }{ 4 }}\]

Answer 5

Oh, I know that I can make this into 9^(70/56),
but is there any other possible way?

Answer 6

Oh, is that the only way I can solve this type of question? By making it into exponents?
Is it possible for me to rewrite it again as
\[\sqrt[4]{9^{5}}\]
and somehow solve like that?

Answer 7

Yeah, you should be able to intuitively change the forms of radicals into exponents.

Answer 8

And yes, you can rewrite it like that.

Answer 9

What do you mean by "solve this problem?"

There's nothing here to solve. I think you mean "simplify," not "solve."\[\sqrt[56]{9^{70}}=9^{\frac{ 70 }{ 56 }}=9^{\frac{ 5 }{ 4 }}\]

Answer 10

You could re-write this last result:\[9^{\frac{ 5 }{ 4 }}=[3^2]^{\frac{ 5 }{ 4 }}\]

Answer 11

This can be reduced. Try it.

Answer 12

In order to further reduce it, you can re-write it as\[\Large 9^\frac{ 4 }{ 4 }*9^\frac{ 1 }{ 4 }=9\sqrt{3}\]

Answer 13

Alright. We multiply 2 and 5/4 correct?
We should get 10/4=5/2 as the exponent.
So...
\[3^{5/2}\]

Answer 14

@steve816 how did you get that?

Answer 15

Isn't it a^n times a^m?

Answer 16

Nope, I think you are thinking of\[\Large (a^n)^m=a^{n*m}\]

Answer 17

\[a ^{n}* a ^{m} = a^{n+m}\]

Answer 18

Ohh wait, my bad

Answer 19

You're correct about that.

Answer 20

How did you get \[\Large 9\sqrt{3}\]

Answer 21

\[\sqrt[56]{9^{70}}\\
=\sqrt[56]{(3^2)^{70}}\\
=\sqrt[56]{3^{140}}\\
= 3^{140/56}\\
=3^{112/56}\times3^{28/56}\\
=3^2\times3^{1/2}
\]

Answer 22

I'm still a little confused on how that results in 9 radical 3. But i'll just ask my teacher for a little more clarification. Thanks @steve816 @mathmale and @UnkleRhaukus

Answer 23

\[\sqrt[56]{9^{70}}\\
=\sqrt[56]{(3^2)^{70}}\\
=\sqrt[56]{3^{140}}\\
=\sqrt[56]{3^{112}\times3^{28}}\\
=\sqrt[56]{3^{112}}\times\sqrt[56]{3^{28}}\\
=3^{112/56}\times3^{28/56}\\
=3^{2}\times3^{1/2}\\
=9\times\sqrt3\]