N-RN.2 (Real Number System) - Dealing With Radicals

In a recent post, we explored evaluating expressions with rational exponents in them, but there's more to Common Core Standard N-RN.2. Don't worry, some of it's easier to deal with than what we already tackled.

These are the items listed below the standard, at least according to the IXL website, where I found the list:

• Simplify radical expressions (Algebra 1 - EE.1)
• Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)
• Multiply radical expressions (Algebra 1 - EE.3)
• Add and subtract radical expressions (Algebra 1 - EE.4)
• Simplify radical expressions using the distributive property (Algebra 1 - EE.5)

Is there anything else to deal with? I don't know. Dealing with rational exponents isn't in this list, and yet I think that they might be encountered before Algebra 2. FYI, the notation "EE" stands for Expressions and Equations, which allows me to once again state that "Expressions don't have equal signs and are evaluated, and Equations do have equal signs and are solved."

Simplifying radical expressions is not a difficult task -- as long as you know that it does NOT mean pushing buttons on your calculator and coming up with an approximate decimal equivalent to 8 or 12 or 15 decimal places. Simplifying a radical is similar to reducing a fraction to its lowest terms. It makes it easier to deal with for computations (particularly adding and subtracting, when the radicals have to be "like terms") and comparisons. If the only thing you're planning to do with a radical number is square it, then, yes, simplifying it is a bigger waste of time than converting an improper fraction into a mixed number when it's only going to be used for slope.

There is a very straightforward method of simplifying square roots, but it seems to mystify some of my students who, apparently, never grasped the concept of what a square root (or a perfect square) was in the first place. They memorize steps, but uncertainty about the order causes them to mess up at the very end, removing radical signs from irrational numbers or leaving them in after taking a square root. (For example, they'll write that the square root of nine = the square root of three, instead of three.)

The simplest method involves finding the largest perfect square which is a factor of the radicand (i.e., the number under the radical sign). If it isn't the largest perfect square, then the radical hasn't been fully simplified. An example:

One problem my students face is not understanding the concept of a perfect square, so instead of 25 and 2, then use 5 and 10. After that, they're stuck, or they just decide, for example, that the square root of 5 is the same as 5 without the radical sign.

Because of this, I tried a different approach, using factor trees. They remembered doing them in middle school, and actually liked using them again. (You see, your teacher was right! You are using them again!) The example looked something like this instead:

After they have the prime factorization under the radical, I have them circle the pairs of numbers, cross them out and write one factor outside the radical. This has two downsides to it: first, if the number has a lot of factors, there will be a lot of extra work (but at least they will know, for certain, that they simplified their answer); second, if they don't complete the problem, they basically just drew a factor tree, which looks kinds childish and silly from a high school student.

Multiplying, Adding and Subtracting Radicals

Multiplying two radicals is as simple as multiplying two fractions. Just multiply the numbers under the radicand. For instance, radical 7 times radical 10 equals radical 70. If the number can be simplified, do it, according to the rules above. Obviously, if you square a radical, such as radical 6 times radical 6, the radical symbol goes away. In this case, you get radical 36, which is just 6.

As mentioned above, if you want to add or subtract radicals, they have to be alike. You can't add or subtract the following the way they are:

They aren't alike. It's like two to add 5² + 4² and getting 9². (In other words, you don't.)

But if you simplify the radicals, how to combine them becomes much clearer:

Finally, there is Division, but I'll save that for another column because of the standard, above, Simplify radical expressions by rationalizing the denominator.