Telling the Truth About Square Roots

Good Morning. This is the second post in a series about topics that have traditionally not been handled honestly in Mathematics classroom. The first post was about adding and subtracting fractions.  In this post I would like to address two aspects of square roots that are often treated with less precision than they ought to be; square roots of negative numbers and simplifying square roots.

To better understand both of these issues, it is important to have a clear understanding of what is meant when we say that a is the square root of b. We mean that a times a is equal to b. More precisely,

.

Square Roots of Negative Numbers

When students first learn about square roots, they generally have only worked with the set of Real numbers; natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The collection of all of these numbers is called the Real numbers. But "real" in this sense has nothing to do with these numbers existing and other numbers not existing. As long as they are only working within the set of real numbers, the statement, "you cannot take the square root of a negative number" is technically correct. A more accurate way to express the same problem would be to say that, "you cannot take the square root of a negative number and get a real number as a result." 

For those of you that don't remember, the number i is defined such that

or.

Using the property that 

,

we can then say that

,

which is NOT a Real number. It is what mathematicians call an Imaginary number. Which isn't the best name, but keep in mind that the set of numbers to which these numbers DO NOT belong is called the Real numbers and the idea was to communicate the fact that these numbers did not belong to the set of Real numbers.

What's more, this result is in line with what we mean by the square root. That is, 

.

Am I suggesting that the teacher dive head first into a lesson on the imaginary unit and complex numbers? No. But at least tease the students with their existence. Let the students know that there is more to come. Give them a reason to explore mathematics on their own.

Simplifying Square Roots

A perfect square is a number whose square root is an integer. The number 25 is a perfect square because the principal square root of 25 is 5. Whereas, 50 is not a perfect square since the square root of 50 is a decimal a little bit bigger than 7.

Right around the time Algebra students start solving quadratic equations using the Quadratic formula, they also learn about simplifying square roots. The basic idea behind simplifying square roots is to take the square root of a number that is NOT a perfect square and express it in a simplified form using the property stated above. For example,

.

Likewise,

.

When this skill is taught, it is generally taught using the aforementioned property along with the dictum, "find the largest perfect square that goes into the radicand." This is then repeated when students learn how to simplify roots with greater index, "find the largest perfect cube that goes into the radicand." While students may be somewhat familiar with small perfect squares, they are less familiar with larger squares and perfect cubes. So simplifying radicals becomes increasingly difficult when larger squares and cubes are needed.

Instead, use the definition of square root (or cube root, or fourth root, etc.) and prime factorization, since these better address what is meant by the square root. To simplify the square root of some number b, presumably not a perfect square, but even if it is, we need to take its prime factorization and split the prime factors into two identical groups, each comprising the factorization of the sought after a.

For example, suppose we wanted to simplify

.

A quick look tells us that 756 is even and so we can divide by 2 to get started with the prime factorization. 756 = 2 x 2 x 3 x 3 x 3 x 7. The goal is to split these factors into two identical groups. This is simple when a factor shows up an even number of times. If a factor shows up an odd number of times, we will split as many as we can until there is only one left. At that point, the leftover factor gets split by taking its square root. In this example, each group can have its own 2 and its own 3. But there is a leftover 3 and a leftover 7. The square root is just the product of the factors from one of the two groups. In this case,

.

As I indicated above, this process can be extended to find higher indexed roots. Just make sure to split the prime factors into three groups for cube roots, four groups for fourth roots, and so on. This process also works when there are variables in the radicand. 

What I believe makes this method better than the "largest square" method usually taught is that it makes better use of the definition of the square root and deepens the students' understanding of this critical concept. Whereas the largest square method, which is pretty close to "guess-and-check," only deepens their understanding of one particular property.