# Rewriting Expressions with Radicals to Expressions with Rational Exponents and Simplifying – HSN.RN.A.2

Let’s say you had the following problem √(x3)and you wanted to change that to an expression with a rational exponent.

Since there is no number written in the radical sign, we know that number is 2, because we are dealing with a square root. That number will be the bottom, or denominator, of the fractional (rational) exponent. The power that x is raised to, in this case 3, is the top, or numerator, of the fractional exponent. So what we get is x3/2.

If you had a cube root or a fourth root, like ∛ or ∜ then you would just use the number inside the “crook” of the radical as your denominator.

Now let’s say you had ∜(168), well that would be 168/4 which would equal 162 which is 256. You would want to make sure to simplify it all the way.

A more complex problem would be ∛(164). In that case you would split the problem up into ∛(163) times ∛(16). You could convert ∛(163) to 163/3 which simplifies to just 16. Then you would simplify it to: 16∛(16). But wait! 16 is 23 times 2. You can “pull out” the 2 from the cube root sign multiply it by the 16 out front to get 32, and leave the other 2 under the cube root. So your complete simplified answer would be 32∛(2).

It’s as simple as that!