Question

What is the cube root of `27a^12`?

Answer 1

The cube root of `27a^12` is `color(red)(3a^4)`

Explanation:

Let's call the term we are looking for `n`. We can then write this problem as:

`n = root(3)(27a^12)`

And, because `root(color(red)(n))(x) = x^(1/color(red)(n))` we can then rewrite it as:

`n = (27a^12)^(1/3)`

Next, we can rewrite `27` as:

`n = (3^3a^12)^(1/3)`

Now, we can use the rule of exponents to eliminate the exponent outside the parenthesis: `(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))`

`n = (3^color(red)(3)a^color(red)(12))^color(blue)(1/3)`

`n = 3^(color(red)(3)xxcolor(blue)(1/3))a^(color(red)(12)xxcolor(blue)(1/3))`

`n = 3^(3/3)a^(12/3)`

`n = 3^1a^4`

And using this rule of exponents we can complete the solution:

`a^color(red)(1) = a`

`n = 3^color(red)(1)a^4`

`n = 3a^4`