Just as the square and the square root are inverses and “undo” each other, the power of exponents help us to simplify radicals.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
As we saw in our previous lesson on Square Roots, the radical notation is comprised of the radical symbol, the radicand or the expression under the radical sign and the index which determines the type of radical we are dealing with.
But sometimes it isn’t easy to work within the confines of the radical notation, and it is better to transform the radical into a rational exponent, and as we progress through the lesson I will evaluate and simplify each radical using two different methods: rational exponents and as I like to call it, the buddy system.
What’s the buddy system?
How to Express Radicals as Rational Exponents
If we are taking a square root, our prime factorization must have two numbers that are identical. If we are taking a cube root, then we need three numbers that are identical. And if we are taking a 4th root, we need four numbers that are identical…. we need buddies!
We will begin our lesson with learning how to convert from radicals to rational exponents and back again. In doing so, we will utilize the skills we gained when we studied exponent properties, such as power of a power.
Next, we will understand the Principal nth root property, and discover that there are instances when we can find the nth root of a negative number. As long as value of the index is a odd, the radicand can be positive or negative.
Why?
Because, if we raise a negative number to an odd exponent, we get a negative number therefore, the inverse operation which is the nth root will similarly give us a negative value, as Paul’s Online Notes accurately states.
Together we will look at countless examples, showing both methods for simplifying so that you can feel comfortable in simplifying any root.
Rational Exponents (How-To) – Video
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