Teaching Square Roots

Radical symbol:  $\sqrt{\text{radicand goes here}}$ 

Key Question:  What are square roots?

First Real Issue: The second way to think of a square root is as a number, that may or may not have a simpler way of being written.  The square root of 25 can be read as the number squared that is 25.  The square root of 9 is the number squared that is 9.

The number squared that is the radicand.

or

What squared is the radicand?

This second way is more powerful because it offers students an avenue of approach to simplify non-obvious radical expressions.

Consider $\sqrt{48}$.  The radicand is not a perfect square.  Thus, this will be our first example of an irrational number.  (A number that cannot be written as a ratio of two integers.)  At some point students should understand that an irrational number cannot be accurately written in decimal notation.  The best way of writing this type of irrational number is with radical notation.

The key word when reading this radical expression with our second understanding is “SQUARED.”  We are looking for perfect squares, or square numbers.  If the radicand has factors that are square numbers, then it can be rewritten or simplified.  (Student need to understand that contextually, the word simplified in the context of square roots means to rewrite the expression so that the radicand does NOT contain factors that are square numbers.)

Our radicand here, 48, has multiple square factors, not considering 1.  We have 4 and 12, with 12 have 3 and 4, and also 16 and 3.  Factors are multiplicative parts of a larger number, numbers whose product is the original number.  So, to factor a number is to break the number into its multiplicative portions.

Key Questions:  (1) Why is it best practice, when simplifying a square root, to factor the radicand and look for the largest perfect square? (2)  List everything you know about the square root of x.

Arithmetic of Irrationals and Checking: In order for students to begin to understand arithmetic with rational and irrational numbers they need to understand that $\sqrt{16}\times \sqrt{3},$ and are the same, but $\sqrt{16+3}\ne \sqrt{16}+\sqrt{3}.$ Square roots are dealing with repeated multiplication, and the order in which numbers are multiplied does not affect the product.  However, multiplication and addition cannot be interchanged.

This is easily seen when dealing with the product and sum of the square root of perfect squares.  For example, with evaluation students can see that $\sqrt{4}\times \sqrt{9}=\sqrt{4\times 9},\sqrt{4}+\sqrt{9}\ne \sqrt{4+9}.$ 

Since $\sqrt{48}=\sqrt{16}\times \sqrt{3},$ and because the square root of sixteen is four, $\sqrt{48}=4\times \sqrt{3}.$ However, in Algebra we rarely use the symbol × to denote multiplication because it is too easily confused with the variable x.  So instead, we place “things,” next to one another to denote multiplication.  Thus, we write $\sqrt{48}=4\sqrt{3}.$ 

This is fully simplified because three, the radicand, does not contain a square number as a factor.  Further, three is a prime number, because it only has two factors, one and itself. 

It is very important that students see that “four $–$ root three,” squared is forty-eight.  This involves arithmetic between rational and irrational numbers, which is something they likely do not understand.

They need to understand that $4\sqrt{3}$ cannot be written more accurately.  A calculator can give a decimal approximation, but the problem with approximations is the inaccuracies in approximations become larger as more computation is done with them.  Thus, for accuracy, it is often best to not approximate a number like the square root of three.

Because $4\sqrt{3}$ cannot be simplified, then ${\left(4\sqrt{3}\right)}^2,$ needs some special attention. 

${\left(4\sqrt{3}\right)}^2=4\sqrt{3}\times 4\sqrt{3}$ 

Of course $4\sqrt{3}$ is the same as $4\times \sqrt{3}.$  So,

$4\sqrt{3}\times 4\sqrt{3}=4\times \sqrt{3}\times 4\times \sqrt{3}.$ 

Make sure students understand that this is not a step, but simply a way to consider how to manipulate such an expression.

Because we can change the order of multiplication, and we can rewrite 4×4 as 16,

$4\times \sqrt{3}\times 4\times \sqrt{3}=4\times 4\times \sqrt{3}\times \sqrt{3}.$ 

Eventually, when exponential notation for square roots is introduced, students will understand another reason why square root of three times itself is the square root of nine. 

$16\times \sqrt{9}.$

The square root of nine is of course three.

$16\times 3=48.$

Students should know learn how to multiply irrational and rational numbers together.  Having them confirm their simplification of a radical expression is a great way to promote this understanding and skill. 

Here we have seen that $\sqrt{48}=4\sqrt{3},$ because ${\left(4\sqrt{3}\right)}^2=48.$ 

This aligns with our second way of viewing square roots, “What squared is the radicand?”

Key Question: If $\sqrt{a{b}^3}=k,\text{}$ what is $k^2?$ 

More Arithmetic with Irrationals: The last issue we are going to address with square roots is addition and subtraction.  To do so, it is advisable to have students discuss why they can add or subtract variables like x and 3x, or 2ab and ab, but they cannot add or subtract unlike terms.  The reason, “Because they’re not like terms,” is unacceptable here. 

The questions are:

Why is it incorrect to add or subtract terms that are unlike? 

Why can unlike terms be multiplied?

The answers deal with the order of operations and the relationship between addition and multiplication.  We can add 3x and 5x, because 3x is really x + x + x, and 5x is x + x + x + x + x.  So we are counting the same thing, so the total number of that thing grows.  However, 3x and 5y are different “things.”  While there are 8 total things, they’re not all the same. 

Multiplication is different.  It is repeated addition.  If we have 3x times 5y, that would be 3x groups of 5y, giving us 15xy.    Spend time chasing this with students, making sure they really understand as it will play an important role as more Algebra is introduced.

This all explains, with understanding, why $3\sqrt{2}+5\sqrt{2}=8\sqrt{2},$ and why $3\sqrt{2}+5\sqrt{3},$ cannot be added.  Further, $3\sqrt{2}\times 5\sqrt{2}=15\sqrt{4}\text{(not simplified)},$ and $3\sqrt{2}\times 5\sqrt{3}=15\sqrt{6}.$

Summary:  When teaching square roots it is easy to resort to simple procedure, and allow students to revert to the simple procedures (that have no conceptual foundation) in order to get the right answers.  Your job is to challenge their understanding of these procedures, help students root their understanding in the concept. 

At the simplest, a square root is a way to write a number.  If taught through discussion and questioning, students can come away with a solid foundation in the subsets of the Real Numbers, factoring, what prime numbers are, how the order of operations respects how multiplication is repeated addition, as well as the students’ mathematical literacy.