Simplifying $9^{3/4}$, I get $3\sqrt[4]{9}$, but that's not the answer. Why?

Question

I am trying to simplify:

$9^\frac{3}{4}$

So this is what I did:

$9^\frac{3}{4} = \sqrt[4]{9^3}$

$\sqrt[4]{3*3*3*3*3*3}$

$3\sqrt[4]{3*3}$

$3\sqrt[4]{9}$

$3\sqrt[4]{3^2}$

I don't see how I can simplify this even more, however the answer I provided is incorrect. How can I simplify this even more?

Answer 1

We know that $9=3^2$ .So, $$\sqrt [4]{9^3} =\sqrt [4]{(3^2)^3} =\sqrt [4]{3^2*3^2*3^2} =\sqrt {3*3*3*3*3*3} $$ After this you have proceeded correctly. You can simplify the last step as: $$ 3\sqrt [4]{3^2} =3\times 3^{2/4} =3\times 3^{1/2} =3\sqrt {3} $$ Hope it helps.

Answer 2

Your answer is correct, but I would leave it as $3\sqrt[4]{9}$.

EDIT: In the spirit of other answers, I might try this instead:

$$9^{3/4}=(3^2)^{3/4}=3^{(2)(3/4)}=3^{6/4}=3^{3/2}=3^{1+1/2}=3\cdot3^{1/2}=3\sqrt{3} $$

Answer 3

Simplify it this way:

$ 9^\frac{3}{4} \\ = (3^2)^\frac{3}{4}\\ = 3^\frac{2\times 3}{4}\\ = 3^\frac{3}{2}\\ = 3^{\left(1 + \frac{1}{2}\right)}\\ = 3 \times 3^\frac{1}{2}\\ = 3 \sqrt{3} $