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The Illini method for simplifying a radical

February 7, 2008, 9:12 pm

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say \(\sqrt{50}\). Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

\(\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & \end{array}\)

Now look for a prime that divides the lower-left term, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

\(\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 & \end{array}\)

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

\(\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 & 2 \\ 1 & \end{array}\)

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5′s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with \(5 \sqrt{2}\).

Here’s another example with a larger number, \(\sqrt{2112}\):

\(\begin{array}{r|r} \sqrt{2112} & 2 \\ 1056 & 2 \\ 528 & 2 \\ 264 & 2 \\ 132 & 2 \\ 66 & 2 \\ 33 & 3 \\ 11 & 11 \\ 1 & \end{array} \)

There are three groups of 2′s, so outside the final radical we’ll put \(2 \cdot 2 \cdot 2 = 8\). And the 3 and 11 are by themselves, so under the radical we put 33. Hence \(\sqrt{2112} = 8 \sqrt{33}\).

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

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