High School: Number and Quantity
The Real Number System HSN-RN.A.1
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5⅓ to be the cube root of 5 because we want (5⅓)3 = 5(⅓)3 to hold, so (5⅓)3 must equal 5.
Your students were in fourth or fifth grade when they first learned about exponents. They thought they were mathematical geniuses because they knew that 5 to the second power was 25. And they were pretty sure that was basically all there was to know.
Then someone explained that anything to the zero power is one and their heads almost exploded. Well, hopefully they'll hang on to their heads this time, because they're about to learn a whole lot more about exponents.
Students should know that when we multiply powers of the same base, the exponents are added together. No surprise there. So 9½ × 9½ should be the same as 9½ + ½ which is 91 (or just 9).
But wait! If we multiply 3 by itself, we also get 9. So 9½ must equal 3!
Are the students confused? Here are a few simple rules for them to follow.
- You're allowed to have exponents that are fractions. It's really okay.
- The denominator of the fraction is the root. So a denominator of 2 means a square root, a denominator of 3 means a cube root, and a denominator of 10 means the tenth root. (Make sure they know you aren't making this up!)
- The numerator of the fraction is the power. So a number to the two-thirds power is the cube root of the number squared.
- It doesn't matter which we do first. If we want to evaluate 8⅔, we have two choices: square 8 and then take the cube root, or take the cube root of 8 and then square it. We'll get the same answer either way. Most people prefer the second way, since it keeps the numbers smaller.
- If your students are using a scientific calculator, the combination of the exponent and fraction keys will allow them to raise numbers to fractional exponents. Still, they should really do the exercises without a calculator. That way, if their calculator is at home on the kitchen table, they'll still survive math class.