## Square Roots

The lessons on this page cover square roots in their entirety.  Part one teaches you the basics of square roots, what they are, where the come from, and what they mean.  Part two teaches you how to simplify square roots.  Part three covers arithmetic of square roots, and part four continues through with more complicated arithmetic, including division.

If you’re a teacher that is interested in introducing square roots conceptually, and you don’t want to re-invent the wheel, the following page might be helpful:  Teaching Square Roots Conceptually.

The first treatment of square roots is all about establishing a quality conceptual understanding of what square root numbers are.

sqrtspt2

Square Roots
(For students)

Part 1

Introduction: The following is the outline of a lesson, the key ideas covered in sequence.  It is best to write your own lessons, when possible, and this is a good guide for what to cover in order.  As far as pace and practice problems, those are left to you.  You will find practice problems at the end that you can use to check for understanding or as homework or quizzes.

Square Roots Ask a Question:  What number squared is equal to the radicand?  The radicand is the number inside the square root symbol (radical).  This expression asks, what number times itself (squared) is 11?

11$\sqrt{11}$

This is a number.  It is not 11.  It turns out this number is irrational and we can never actually write what it is more accurately than this.

Big Idea:  The area of a square is calculated by squaring a side (multiplying it by itself).  Since all sides of a square are equal, this about as easy of an area to calculate as possible.  A square root is giving us the area of a square and asking us to find out how long a side is.

42$\overline{)42}$

For example, this square has an area of forty-two.  Instead of writing out the question, “How long is the side of a square whose area is forty-two?” we simply write, 42.$\sqrt{42}.$

The majority of the confusion with square roots comes back to this definition of what a square root is.  To make it as clear as possible, please consider the following table.

 English Math How long is the side of a square that has an area of 100? 100−−−√$\sqrt{100}$ How long is the side of a square that has an area of 10? 10−−√$\sqrt{10}$

These two numbers were chosen because students inevitably write 100=10.$\sqrt{100}=\sqrt{10}.$

 English Math Answer How long is the side of a square that has an area of 100? 100−−−√$\sqrt{100}$ 10 How long is the side of a square that has an area of 10? 10−−√$\sqrt{10}$ 3.162277660168379…

Key Knowledge:  In order to be proficient with square roots we need to know about perfect squares.  A perfect square is a number that is the product of a number squared.  Sixteen is a perfect because four times four is sixteen.

The reason you need to know perfect squares is because square roots are asking for numbers squared that equal the radicand.  So if the radicand is a perfect square, we have an easy ‘get,’ that is, simplification.

For example, since 42 = 16, and the square root of sixteen (16)$\left(\sqrt{16}\right)$ asks for what number squared is 16, the answer is just four.

Let’s take a look at the first twenty perfect squares and what number has been squared to arrive at the perfect square, which we will call the parent.

 Perfect Square “Parent” 1 1 4 2 9 3 16 4 25 5 36 6 49 7 64 8 81 9 100 10 121 11 144 12 169 13 196 14 225 15 256 16 289 17 324 18 361 19 400 20

You should recognize these numbers as perfect squares as that is a key piece of knowledge required!

Pro-Tip:  When dealing with square roots it is wise to have a list of perfect squares handy to help you familiarize yourself with them.

How to Simplify a Square Root:  To simplify a square root all you do is answer the question it is asking.

The best way to go about that is to see if the radicand is a perfect square.  If so, then just answer the question.  For example:

Simplify 256$\sqrt{256}$

Since this is asking, “What number squared is 256?” and 256 is a perfect square, 162, the answer to the question is just 16.

256=16$\sqrt{256}=16$

What if we had something like this:

x2$\sqrt{{x}^{2}}$

If you’re confused by this, revert back to the question it is asking.  This is asking, “What squared is x $–$ squared?”  All you have to do is answer it.

x2=x$\sqrt{{x}^{2}}=x$

What if the radicand was not a perfect square?

If you end up with an ugly square root, like 48,$\sqrt{48},$ all you have to do is factor the radicand to find the largest perfect square.

List all factors, not just the prime factors.  In fact, the prime factors are of little use because prime numbers are not perfect squares.  And again, we are looking for perfect squares because they help us answer the question posed by the square root.

48

1, 48

2, 24

3, 16

4, 12

6, 8

Pro-Tip:  When factoring, do not skip around.  Check divisibility by all of the numbers in order until you get a turn around.  For example, after 6, check 7.  Seven doesn’t divide into 48, but 8 does.  Eight times six is forty eight, but you already have that pair.  That’s how you know you’re done!

In our list we need to find the largest perfect square.  While four is a perfect square, sixteen is larger.  So we need to use three and sixteen like shown below.

48=163$\sqrt{48}=\sqrt{16}\cdot \sqrt{3}$

The square root of three is irrational (square roots of prime numbers are all irrational), but the square root of sixteen is four.  So rewriting this we get:

48=16348=43$\begin{array}{l}\sqrt{48}=\sqrt{16}\cdot \sqrt{3}\\ \sqrt{48}=4\cdot \sqrt{3}\end{array}$

See Note 1 and Note 2 below for an explanation of why the above works.

Fact:

m=k if k2=m.

That means that 48=43, if (43)2=48. Let’s see if it is true.

(43)2=43×43${\left(4\sqrt{3}\right)}^{2}=4\sqrt{3}×4\sqrt{3}$

Because we can change the order in which we multiply, we can rearrange this and multiply the rational numbers together first and the irrational numbers together first.

43×43=4×4×3×3$4\sqrt{3}×4\sqrt{3}=4×4×\sqrt{3}×\sqrt{3}$

The square root of three times itself is the square root of nine.

4×4×3×3=16×9$4×4×\sqrt{3}×\sqrt{3}=16×\sqrt{9}$

The square root of nine asks, what squared is nine.  The answer to that is three.

16×9=16×3$16×\sqrt{9}=16×3$

So,

48=43 because (43)2=48.

Note 1:  43$4\sqrt{3}$ cannot be simplified because the square root of three is irrational.  That means we cannot write it more accurately than this.  Also, the product of rational number and an irrational number is irrational.  So, 43$4\sqrt{3}$ is just written as “four root three.”

Note 2:  We can separate square roots into the product of two different square roots like this:

75=253$\sqrt{75}=\sqrt{25\cdot 3}$ figure a.

or

75=253$\sqrt{75}=\sqrt{25}\cdot \sqrt{3}$ figure b.

If we consider the question being asked, what number squared is seventy five, we can see why this works.  What number squared is seventy five is the same as what number squared is twenty five times three,” (figure a).  The number squared that is twenty five times the number squared that is three is the same as the number times itself that is twenty five times three.

For example:  64=8 because 82=64.

But also:  64=416$\sqrt{64}=\sqrt{4}\cdot \sqrt{16}$

And this simplifies to:

64=24=8$\sqrt{64}=2\cdot 4=8$

What we will see in a future section is that square roots are actually exponents, exponents are repeated multiplication and the order in which you multiply does not matter.  This allows us to manipulate square root expressions in such a fashion.

Let us work through two examples.  Before we do, let us define what simplify means in the context of square roots.  Simplify with square roots means that the radicand does not contain a factor that is a perfect square and that all terms are multiplied together.

Simplify: 98$9\sqrt{8}$

What is the nine doing with the square root of eight?  It is multiplying by it.  We cannot carry out that operation.  However, eight, the radicand, does contain a perfect square, four. Do not allow the fact that 9 is also a perfect square confuse you. This is just 9, as in 1, 2, 3, 4, 5, 6, 7, 8, 9.  The square root of eight cannot be counted.  It is asking a question, remember?

98=942$9\sqrt{8}=9\sqrt{4}\cdot \sqrt{2}$

Pro-Tip:  When rewriting radical expressions (square roots), write the perfect square first as it is easier to manipulate (you won’t mess up as easily).

942922182$\begin{array}{l}9\sqrt{4}\cdot \sqrt{2}\\ 9\cdot 2\cdot \sqrt{2}\\ 18\sqrt{2}\end{array}$

Example 2

Simplify 832x4$8\cdot \sqrt{\frac{32}{{x}^{4}}}$

The eight is multiplying with the radical expression.  Just like we could separate the multiplication of square roots, we can also separate the division, provided it is written as multiplication by the reciprocal.  So, let’s consider these separately, to break this down into smaller pieces that are easier to manage.

832x4=8×32x4$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{\sqrt{32}}{\sqrt{{x}^{4}}}$

Let’s factor each square root, looking for a perfect square.  Note that x2 times x2 is x4

832x4=8×16×2x4$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{\sqrt{16}×\sqrt{2}}{\sqrt{{x}^{4}}}$

832x4=8×4×2x2$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{4×\sqrt{2}}{{x}^{2}}$

Notice that 8 is a fraction 8/1.

832x4=81×4×2x2$8\cdot \sqrt{\frac{32}{{x}^{4}}}=\frac{8}{1}×\frac{4×\sqrt{2}}{{x}^{2}}$

Multiplication of fractions is easy as π.

832x4=322x2$8\cdot \sqrt{\frac{32}{{x}^{4}}}=\frac{32\sqrt{2}}{{x}^{2}}$

Summary: Square roots ask a question:  What number squared is the radicand?  This comes from the area of a square.  Given the area of a square, how long is the side?

To answer the question you factor the radicand and find the largest perfect square.

## Introduction to Square Roots

Note:  These lessons on square roots are designed to expose some misconceptions students hold regarding arithmetic with irrational numbers.  This misconception creates problems when they begin to manipulate algebraic expressions, use more complicated formulas and solve equations.

These misconceptions are addressed as they naturally occur with square roots.  The concepts learned by students in these two lessons are:

• How square roots are an operation
• How a square root of a number is, itself, just a number
• Why a square root number can equal another number
• How square roots are the most efficient way of writing many irrational numbers
• How square roots can be read as a question or a statement

The procedures students will learn in these two lesson are:

• How to multiply irrational and rational numbers
• How to simplify the square root of a square number
• How to simplify the square root of a non-square composite number
• How to verify if a simplified square root is correct

Big Idea

• Square roots are more than just an operation. They can just be a way of writing numbers.  Sometimes they can be rewritten without changing the exact value, but sometimes are the most accurate way of writing an irrational number.
• Square roots can be read two ways:
• If the area of a square is the radicand, how long is the side?
• What squared is the radicand?

Key Knowledge

• For students to be proficient with square roots they must know:
• Perfect Squares (square numbers)
• Prime numbers
• Factoring

Pro-Tip

(for students)

• When simplifying square roots, students should factor the radicand to find the largest perfect square. Then, they rewrite the square root of the perfect square with the product of the irrational portion of the number.
• Know all of the square numbers up to 400.

You can access the lesson guide in the Packet.

### Square Roots Part 1

With a $1.00 purchase of the packet you help support this website. What you get in this packet is: • Topic Reference Sheet • Lesson Guide • PowerPoint • Assignment That’s just a quarter per resource! What a steal! If for any reason you are unsatisfied with your purchase, your money will be fully refunded and you get to keep the materials. Click the PayPal icon below to safely and securely complete your transaction. In the second part of square roots we deal with simplifying square root numbers whose radicands are not square numbers. Introduction to Square Roots Arithmetic Note: These lessons on square roots are designed to expose some misconceptions students hold regarding arithmetic with irrational numbers. This misconception creates problems when they begin to manipulate algebraic expressions, use more complicated formulas and solve equations. These misconceptions are addressed as they naturally occur with square roots. The concepts learned by students in these two lessons are: • How square roots are an operation • How a square root of a number is, itself, just a number • Why a square root number can equal another number • How square roots are the most efficient way of writing many irrational numbers • How square roots can be read as a question or a statement The procedures students will learn in these two lesson are: • How to multiply irrational and rational numbers • How to simplify the square root of a square number • How to simplify the square root of a non-square composite number • How to verify if a simplified square root is correct Big Idea • Square roots are more than just an operation. They can just be a way of writing numbers. Sometimes they can be rewritten without changing the exact value, but sometimes are the most accurate way of writing an irrational number. • Square roots can be read two ways: • If the area of a square is the radicand, how long is the side? • What squared is the radicand? Key Knowledge • For students to be proficient with square roots they must know: • Perfect Squares (square numbers) • Prime numbers • Factoring Pro-Tip (for students) • When simplifying square roots, students should factor the radicand to find the largest perfect square. Then, they rewrite the square root of the perfect square with the product of the irrational portion of the number. Lesson Guide This is day two of Square Roots for High School. Included are an in-class pop quiz, and a homework assignment. To download the PowerPoint,.  Time Notes Slide # 5 Review the pop quiz (if given) and/or homework 2 – 3 2 Introduce the topic … ask a student if they know what composite means. 4 3 The reason that is because 52 = 25 … not because of factors, pairs of primes, or square numbers. This is a slippery concept, but once students lose sight of it, they get very confused in their procedures! 5 10 – 15 Walk them through the process. There is some good vocabulary for them to learn…things they heard and should know, but likely do not. 6 – 8 5 – 10 This will teach them how to multiply a ration and irrational number. They likely do not understand how, or in practical terms, that the product is irrational. They learned to approximate in middle school, and quite likely have confused irrational number arithmetic with rational number arithmetic. (Not the fault of a MS teacher, just a commonly developed misconception.) 9 – 10 10 – 15 Give students four minutes (in silent work) to try as many as possible. Then allow them to help one another. Finally, review mistakes and misconceptions you witnessed as a whole group.NOTE: If this is beyond them, then you’ll need to slow down here. 11 10 Students need to make the connection between factoring numbers and unknown expressions. Conduct these slides with as much questioning as possible to help students make sense of it. 12 – 14 5 Closure 15 Homework 16 With a$2.00 purchase of the packet you help support this website.  What you get in this packet is:

• Topic Reference Sheet
• Lesson Guide
• PowerPoint
• Assignment
If for any reason you are unsatisfied with your purchase, your money will be fully refunded and you get to keep the materials.

Click the PayPal icon below to safely and securely complete your transaction.

Part three of square roots really gets into the concepts behind why arithmetic with square root numbers works.  This is a very powerful section that helps students really understand why what it is they’re doing works.

Mathematical Operations and Square Roots

Part 1

In this section we will see why we can add things like 52+32$5\sqrt{2}+3\sqrt{2}$ but cannot add things like 25+23$2\sqrt{5}+2\sqrt{3}$.  Later we will see how multiplication and division work when radicals (square roots and such) are involved.

Addition and Subtraction:  Addition is just repeated counting.  The expression 52$5\sqrt{2}$ means 2+2+2+2+2$\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}$, and the expression 32 means 2+2+2.  So if we add those two expressions, 52+32,$5\sqrt{2}+3\sqrt{2},$ we get 82$8\sqrt{2}$ .  Subtraction works the same way.

Consider the expression 25+23$2\sqrt{5}+2\sqrt{3}$.  This means 5+5+3+3.$\sqrt{5}+\sqrt{5}+\sqrt{3}+\sqrt{3}.$  The square root of five and the square root of three are different things, so the simplest we can write that sum is 25+23$2\sqrt{5}+2\sqrt{3}$.

A common way to describe when square roots can or cannot be added (or subtracted) is, “If the radicands are the same you add/subtract the number in front.”  This is not a bad rule of thumb, but it treats square roots as something other than numbers.

5×3+4×3=9×3$5×3+4×3=9×3$

The above statement is true.  Five groups of three and four groups of three is nine groups of three.

53+43=93$5\sqrt{3}+4\sqrt{3}=9\sqrt{3}$

The above statement is also true because five groups of the numbers squared that is three, plus four more groups of the same number would be nine groups of that number.

However, the following cannot be combined in such a fashion.

3×8+5×2$3×8+5×2$

While this can be calculated, we cannot add the two terms together because the first portion is three $–$ eights and the second is five $–$ twos.

38+52$3\sqrt{8}+5\sqrt{2}$

The same situation is happening here.

Common Mistake:  The following is obviously wrong.  A student learning this level of math would be highly unlikely to make such a mistake.

7×2+9×2=16×4$7×2+9×2=16×4$

Seven $–$ twos and nine $–$ twos makes a total of sixteen $–$ twos, not sixteen $–$ fours.  You’re adding the number of twos you have together, not the twos themselves.  And yet, this is a common thing done with square roots.

72+92=164$7\sqrt{2}+9\sqrt{2}=16\sqrt{4}$

This is incorrect for the same reason.  The thing you are counting does not change by counting it.

Explanation:  Why can you add 52+32$5\sqrt{2}+3\sqrt{2}$?  Is that a violation of the order of operations (PEMDAS)?  Clearly, the five and square root of two are multiplying, as are the three and the square root of two.  Why does this work?

Multiplication is a short-cut for repeated addition of one particular number.  Since both terms are repeatedly adding the same thing, we can combine them.

But if the things we are repeatedly adding are not the same, we cannot add them together before multiplying.

What About Something Like This:  340990$3\sqrt{40}-9\sqrt{90}$?

Before claiming that this expression cannot be simplified you must make sure the square roots are fully simplified.  It turns out that both of these can be simplified.

340990$3\sqrt{40}-9\sqrt{90}$

34109910$3\cdot \sqrt{4}\cdot \sqrt{10}-9\cdot \sqrt{9}\cdot \sqrt{10}$

The dot symbol for multiplication is written here to remind us that all of these numbers are being multiplied.

34109910$3\cdot \sqrt{4}\cdot \sqrt{10}-9\cdot \sqrt{9}\cdot \sqrt{10}$

32109310$3\cdot 2\cdot \sqrt{10}-9\cdot 3\cdot \sqrt{10}$

6102710$6\sqrt{10}-27\sqrt{10}$

2110$-21\sqrt{10}$

What About Something Like This:  7+7$\sqrt{7+7}$ versus 7+7.$\sqrt{7}+\sqrt{7}.$

Notice that in the first expression there is a group, the radical symbol groups the sevens together.  Since the operation is adding, this becomes:

7+7=14$\sqrt{7+7}=\sqrt{14}$.

Since the square root of fourteen cannot be simplified, we are done.

The other expression becomes:

7+7=27.$\sqrt{7}+\sqrt{7}=2\sqrt{7}.$

Summary:  If the radicals are the same number, the number in front just describes how many of them there are.  You can combine (add/subtract) them if they are the same number.  You are finished when you have combined all of the like terms together and all square roots are simplified.

Introduction to Square Roots

Note:  These lessons on square roots are designed to expose some misconceptions students hold regarding arithmetic with irrational numbers.  This misconception creates problems when they begin to manipulate algebraic expressions, use more complicated formulas and solve equations.

These misconceptions are addressed as they naturally occur with square roots.  The concepts learned by students in these two lessons are:

• How square roots are an operation
• How a square root of a number is, itself, just a number
• Why a square root number can equal another number
• How square roots are the most efficient way of writing many irrational numbers
• How square roots can be read as a question or a statement

The procedures students will learn in these two lesson are:

• How to multiply irrational and rational numbers
• How to simplify the square root of a square number
• How to simplify the square root of a non-square composite number
• How to verify if a simplified square root is correct

Big Idea

• Square roots are more than just an operation. They can just be a way of writing numbers.  Sometimes they can be rewritten without changing the exact value, but sometimes are the most accurate way of writing an irrational number.
• Square roots can be read two ways:
• If the area of a square is the radicand, how long is the side?
• What squared is the radicand?

Key Knowledge

• For students to be proficient with square roots they must know:
• Perfect Squares (square numbers)
• Prime numbers
• Factoring

Pro-Tip

(for students)

• When simplifying square roots, students should factor the radicand to find the largest perfect square. Then, they rewrite the square root of the perfect square with the product of the irrational portion of the number.

This is day three of Square Roots for High School.  To download the PowerPoint, click the icon below.

 Time Notes Slide # 10 Homework Review…conduct a thorough conversation about what students learned, what mistakes they made, and help them shore up misconceptions 2 5 Have students discuss this carefully.  Students want to cite steps as being wrong, but steps come from ideas.  It is the concept that to the articulation of an idea. 3 10 Students will likely say it is wrong without trying the math…which is perfect.  The idea is to get at why we can add before multiplying in some circumstances. 4 – 7 5 – 10 Now that students understand why you can add if there are like terms, it is time to get them to see things like  as terms. 8 – 10 5 Have students try this…do NOT allow them to observe and wait.  They have enough of an idea to make an attempt at this point.  It is only through struggling with, and making mistakes with, new information that they’ll understand.   Have students explain how the new problem on the board is the same as the original… 11 15 This is an appropriate time to split up the work, assigning small groups to a problem and then having them share on the board and then review. 12 – 15 5 Closure 16 Homework 17

Tab Content

With a $4.00 purchase of the packet you help support this website. What you get in this packet is: • Topic Reference Sheet • Lesson Guide • PowerPoint • Two Assignments and Key • Quiz and Key (Real Numbers and Primes) If for any reason you are unsatisfied with your purchase, your money will be fully refunded and you get to keep the materials. Click the PayPal icon below to safely and securely complete your transaction. In part four we discuss rationalizing the denominator and slightly more complicated expression than were seen in part three. This last section is one that is often skipped, but it is worth the extra time. To download the PowerPoint, please click the icon below. With a$2.00 purchase of the packet you help support this website.  What you get in this packet is:

• Topic Reference Sheet
• Lesson Guide
• PowerPoint
• Assignment

If for any reason you are unsatisfied with your purchase, your money will be fully refunded and you get to keep the materials.

Click the PayPal icon below to safely and securely complete your transaction.

Click the button above to download all of the resources published on this page, plus the lesson guides that accompany the PowerPoint presentations.