Question

How do you evaluate and simplify `27^(2/3)`?

Answer 1

See the entire solution process below:

Explanation:

First, we can use these rule for exponents to modify this expression:

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

`27^(2/3) = 27^(color(red)(1/3) xx color(blue)(2)) = (27^(color(red)(1/3)))^color(blue)(2)`

We can now use rule of roots/exponents to modify the expression within parenthesis:

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

`(27^(1/color(red)(3)))^2 = (root(color(red)(3))(27))^2 = (3)^2 = (3 xx 3) = 9`

Answer 2

You can also use logarithmic functions, if you have a logarithm table or at least a calculator that can do log functions, if not complex exponential ones.

Explanation:

Using the transformation of `27^(2/3) = (2/3)* log 27`
we calculate
`(2/3) * 1.431 = 0.9542`
Taking the inverse (anti-log, or exponent) of this value gives us the answer:

`10^(0.9542) = 9`