Question

Rewrite the radical as a rational exponent.

the cube root of 2 to the seventh power

Answer 1

\[\huge x ^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\]

Answer 2

That's it iambatman just said everything I was about to say.
Use that formula and you will be able to get an answer.

Answer 3

so 2 7/3 ?

Answer 4

So how do we make roots as exponents? First off if you have something like \[\LARGE (n^3)^2=(n^3)(n^3)=(n*n*n)*(n*n*n)=n^6\]

That tells us, ahh! It looks like the rule is to multiply exponents if they're something exponentiated raised to an exponent.

So now we look at

\[\LARGE \sqrt{x}^2=x^1\]

So it looks like the x^2 is cancelled out by the square root to equal 1. This is just like the last case, so it must mean that \[\Large \sqrt{x}=x^{1/2}\]

And this makes sense that it will be true for other roots too. Cube root should cancel out with the cube to be 1, so that should also be 1/3 power for cube root.

Answer 5

\[2^{7/3}\]

Answer 6

and to do this reversed?
Rewrite the rational exponent as a radical.

5 to the 3 over 4 power, to the 2 over 3 power

Answer 7

how would I do that?

Answer 8

@WordGEEK @iambatman @Kainui

Answer 9

(53/4) 2/3

Answer 10

What are you trying to ask now?

Answer 11

what would (53/4)2/3 be as a radical ?

Answer 12

\(\Large (5^{ \frac 34})^ \frac 23 \) ?

Answer 13

yes^ :)

Answer 14

Since \(\Large (x^m)^n = x^{mn}\)
Multiply the exponents first. What do you get?

Answer 15

6?

Answer 16

\(\Large \frac 34 * \frac 23\) = ?

Answer 17

1/2 ?

Answer 18

Yup good it equals 1/2 so now just apply the formula that iambatman posted.

Answer 19

Yes. \(\Large x^{\frac 12} = \sqrt{x}\).
Therefore, \(\Large 5^{\frac 12}\) = ?

Answer 20

2\[\sqrt[2]{5}\]

Answer 21

and the one on the outside ?

Answer 22

I don't think that's right

Answer 23

Yes. But 2 is a special case where \(\Large \sqrt [2]{5} = \sqrt{5}\)
Both mean the same. Square root 5 or just radical 5.
\(\Large (5^{ \frac 34})^ \frac 23 = 5^{\frac 34 * \frac 23} = 5^\frac 12 = \sqrt {5}\)

Answer 24

oh okay! now it makes sense! that's one of the answer options. thank you! :)

Answer 25

You are welcome.