# 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 5 to be the cube root of 5 because we want (5)3 = 5(⅓)3 to hold, so (5)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.

1. You're allowed to have exponents that are fractions. It's really okay.
2. 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!)
3. The numerator of the fraction is the power. So a number to the two-thirds power is the cube root of the number squared.
4. It doesn't matter which we do first. If we want to evaluate 8, 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.
5. 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.

#### Drills

1. Evaluate .

10

Remember, the key with fractional exponents is to treat the numerator and denominator separately. The numerator tells you the power, the denominator the root. So  means the square root of 100 to the first power. That's 10 to the first power, or 10. It does not mean half of 100, or 50. But cheer up if you answered (A). It's probably the most common wrong answer out there for a question like this one.

2. Evaluate .

3

The numerator of the exponent is 1, and the denominator is 3. That means we want the cube root of 27 to the first power. The cube root of 27 is 3, since (3)(3)(3) = 27.

3. Evaluate .

64

First, take the square root of 16 because the denominator is 2. (Remember, a denominator of 2 means square root.) So far, we have a 4. Now, we use that numerator of 3 to cube our 4. That gives us 64. Remember, the numerator and denominator take us in different directions.

4. Evaluate .

625

The denominator tells us to take the cube root of 125. So far, that gives us a 5 (since (5)(5)(5) = 125.) Now use the numerator, 4, to determine which power to raise that 5. If we calculate 5 to the fourth power, we should get 625.

5. Which of the following has the largest value?

Yeah, sorry. We just had to give you one problem where you had to find all four answers and compare them. Remember, the rules are that the denominator gives you the root and the numerator gives you the power. With those rules in mind, (A) gives us the square root of 16 to the third power or 64, (B) gives us the square root of 16 to the first power or 8, (C) gives us the cube root of 8 to the fourth power or 16, and (D) gives us the cube root of 125 to the first power, or 5. The winner is (A).

6. Evaluate .

32

Since most people prefer to take the root before they do anything else, we'll do that first. Like we said before, it keeps the numbers smaller. The square root of 4 (square because there's a 2 in the exponent's denominator) is 2. Now we can raise 2 to the power of the exponent's numerator, 5. If we do that, we should get 32, which is (B).

7. Evaluate

81

All we need to do here is take the cube root of 27 (which we already did in question 2, if you remember). That gives us 3. If we raise 3 to the fourth power, we should get a number larger than 27. If we do the math, we get 81 as our answer.

8. Evaluate .

64

At first, this exponent seems a bit weird. (How would you take the first root of a number?) We don't really have to worry about that. Just looking at the exponent, we can simplify  to 2. That means our answer is just 82, or 64 (not 16!).

9. Evaluate .

2

At first,  looks big and scary. Multiplying 4 by itself 100 times and then finding the two hundredth root seems like the very definition of unnecessary. You're right: it is unnecessary. We could just reduce the exponent from  to . They're equal, aren't they? So that means rather than multiplying forever and ever, we can just take the square root of 4 to get 2.

10. Evaluate .

49