Question

How do you write `8^(4/3)` in radical form?

Answer 1

See a solution process below:

Explanation:

First, rewrite the expression as:

`8^(color(red)(4) xx color(blue)(1/3))`

Next, use this rule of exponents to rewrite the expression again:

`x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)9`

`8^(color(red)(4) xx color(blue)(1/3)) = (8^(color(red)(4)))^color(blue)(1/3)`

Now, use this rule for exponents and radicals to write this in radical form:

`x^(1/color(red)(n)) = root(color(red)(n))(x)` or `root(color(red)(n))(x) = x^(1/color(red)(n))`

`(8^4)^(1/color(red)(3)) = root(color(red)(3))(8^4)`

If necessary, we can reduce this further by first rewriting the radical as:

`root(3)(8^3 * 8)`

We can then use this rule for multiplying radicals to simplify the radical:

`root(n)(color(red)(a) * color(blue)(b)) = root(n)(color(red)(a)) * root(n)(color(blue)(b))`

`root(3)(color(red)(8^3) * color(blue)(8)) = root(3)(color(red)(8^3)) * root(3)(color(blue)(8)) = 8root(3)(8)`

Answer 2

The denominator of the exponent tells us the root and the numerator tells us the power.

Explanation:

One way to recall which is which is to think about `n^(1/5)` and `n^5` The second is the same as `x^(5/1)` so the numeraotr gives the power and the denominator gives the root.

`8^(4/3)`

This exponent is a reduced fraction.

If the fraction exponent is already reduced the it doesn't matter what order we use.

`8^(4/3) = root(3)(8^4)` is the same as `root(3)8^4`.

In fact, in the second for, if we notice that we know `root(3)8 = 2`, then we can quickly simplify further.

`root(3)8^4 = (root(3)8)^4 = (2)^4 = 16`

I've tried to show the thought process. It would be fine to write just

`root(3)8^4 = 2^4 = 16`.

Final note

We could also simplify `8^(4/3) = root(3)(8^4)`

(We'd better be able to. I just said they are the same.)

`8^(4/3) = root(3)(8^4) = root(3)(8^3*8)`

`= root(3)(8^3) root(3)8`

`= 8root(3)8`

`= 8*2`

`= 16`