N-RN.2 (Real Number System) - More Rational Exponents

Continuing my post from Tuesday, on the Real Number system and rational exponents, let's move on to Standard N-RN.2, which reads Rewrite expressions involving radicals and rational exponents using the properties of exponents.

There's a lot to consider under this standard, so I'll continue with rational exponents, i.e., fractions. What if we wanted to evaluate an expression like this?

We need to recognize that the radical 5 is the same as 5^1/2, so

The rules for exponents say to multiply the 1/2 and the 4, giving us 5² or 25.

We can take this further. Suppose we had

The cube root is the same as 1/3 power. So

We can evaluate 6³ as 6 * 6 * 6 = 216.

One more example: How would we handle

The fourth root becomes the 1/4 power.

Now we can get a little fancy with and deal with the multiplication of two fractions:

One final note: The answer doesn't always have to be a rational number. You may exchange one rational power for another, one root for a different one. Consider:

Problems could contain any combination of roots and improper fractions, which may or may not have a simple rational answer. But keep the calculator handy just in case you need to know the sixth root of 117,649. Showing your work, of course.