Square Roots Part 1

Note:  Square roots are pretty tricky to teach and learn because the tendency is to seek answer-getting methods.  Patience at the onset, allowing for full development of conceptual understanding is key.  Do not revert to tricks and quick "gets" when first learning square roots.  Always revert back to the question they ask and how you know if you've answered that question.

square roots part 1

Square Roots

Part 1

Introduction: Square roots are consistently among the most misunderstood topics in developmental math. Similar to exponents, students must possess both procedural fluency but also a solid conceptual foundation and the ability to read and understand what square roots mean, in order to be proficient with them. It is often the case that problems with square roots do not lend themselves to a correct first step, but rather, offer many equally viable methods of approach.

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGymaaWcbeaaaaa@3788@

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauIhaeaaca aI0aGaaGOmaaaaaaa@37B8@

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 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaaGOmaaWcbeaakiaac6caaaa@3848@

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGimaiaaicdaaSqabaaaaa@3841@

How long is the side of a square that has an area of 10?

10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGimaaWcbeaaaaa@3787@

 

These two numbers were chosen because students inevitably write 100 = 10 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGimaiaaicdaaSqabaGccqGH9aqpdaGcaaqaaiaaigdacaaI WaaaleqaaOGaaiOlaaaa@3B9D@

English

Math

Answer

How long is the side of a square that has an area of 100?

100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGimaiaaicdaaSqabaaaaa@3841@

10

How long is the side of a square that has an area of 10?

10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGimaaWcbeaaaaa@3787@

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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada GcaaqaaiaaigdacaaI2aaaleqaaaGccaGLOaGaayzkaaaaaa@3920@ 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaaGynaiaaiAdaaSqabaaaaa@384D@

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaaGynaiaaiAdaaSqabaGccqGH9aqpcaaIXaGaaGOnaaaa@3AD8@

What if we had something like this:

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WG4bWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@37ED@

If you’re confused by this, revert back to the question it is asking. This is asking, “What squared is x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ squared?” All you have to do is answer it.

x 2 =x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WG4bWaaWbaaSqabeaacaaIYaaaaaqabaGccqGH9aqpcaWG4baaaa@39FA@

What if the radicand was not a perfect square?

If you end up with an ugly square root, like 48 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaaGioaaWcbeaakiaacYcaaaa@384C@ 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 = 16 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaaGioaaWcbeaakiabg2da9maakaaabaGaaGymaiaaiAdaaSqa baGccqGHflY1daGcaaqaaiaaiodaaSqabaaaaa@3D64@

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 = 16 3 48 =4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaGcaa qaaiaaisdacaaI4aaaleqaaOGaeyypa0deeG+aaaaaaivzKbWdbmaa kaaabaGaaGymaiaaiAdaaSqabaGcpaGaeyyXIC9aaOaaaeaacaaIZa aaleqaaaGcbaWaaOaaaeaacaaI0aGaaGioaaWcbeaakiabg2da98qa caaI0aWdaiabgwSixpaakaaabaGaaG4maaWcbeaaaaaa@4658@

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

Fact:

m =k if  k 2 =m. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGTbaaleqaaOGaeyypa0Jaam4AaiaabccacaqGPbGaaeOzaiaabcca caWGRbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamyBaiaac6caaa a@40AC@

That means that 48 =4 3 , if  ( 4 3 ) 2 =48. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaaGioaaWcbeaakiabg2da9iaaisdadaGcaaqaaiaaiodaaSqa baGccaGGSaGaaGzaVlaabccacaqGPbGaaeOzaiaabccadaqadaqaai aaisdadaGcaaqaaiaaiodaaSqabaaakiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaGccqGH9aqpcaaI0aGaaGioaiaac6caaaa@46EB@ Let’s see if it is true.

( 4 3 ) 2 =4 3 ×4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI0aWaaOaaaeaacaaIZaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaOGaeyypa0JaaGinamaakaaabaGaaG4maaWcbeaaki abgEna0kaaisdadaGcaaqaaiaaiodaaSqabaaaaa@4066@

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.

4 3 ×4 3 =4×4× 3 × 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaka aabaGaaG4maaWcbeaakiabgEna0kaaisdadaGcaaqaaiaaiodaaSqa baGccqGH9aqpcaaI0aGaey41aqRaaGinaiabgEna0oaakaaabaGaaG 4maaWcbeaakiabgEna0oaakaaabaGaaG4maaWcbeaaaaa@45CF@

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

4×4× 3 × 3 =16× 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgE na0kaaisdacqGHxdaTdaGcaaqaaiaaiodaaSqabaGccqGHxdaTdaGc aaqaaiaaiodaaSqabaGccqGH9aqpcaaIXaGaaGOnaiabgEna0oaaka aabaGaaGyoaaWcbeaaaaa@44F2@

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

16× 9 =16×3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dacqGHxdaTdaGcaaqaaiaaiMdaaSqabaGccqGH9aqpcaaIXaGaaGOn aiabgEna0kaaiodaaaa@3FC6@

So,

48 =4 3  because  ( 4 3 ) 2 =48. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaaGioaaWcbeaakiabg2da9iaaisdadaGcaaqaaiaaiodaaSqa baGccaqGGaGaaeOyaiaabwgacaqGJbGaaeyyaiaabwhacaqGZbGaae yzaiaabccadaqadaqaaiaaisdadaGcaaqaaiaaiodaaSqabaaakiaa wIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaI0aGaaG ioaiaac6caaaa@4949@

Note 1: 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaka aabaGaaG4maaWcbeaaaaa@378D@ 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, 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaka aabaGaaG4maaWcbeaaaaa@378D@ 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI3aGaaGynaaWcbeaakiabg2da9maakaaabaGaaGOmaiaaiwdacqGH flY1caaIZaaaleqaaaaa@3D3F@ figure a.

or

75 = 25 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI3aGaaGynaaWcbeaakiabg2da9maakaaabaGaaGOmaiaaiwdaaSqa baGccqGHflY1daGcaaqaaiaaiodaaSqabaaaaa@3D64@ 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  8 2 =64. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI2aGaaGinaaWcbeaakiabg2da9iaaiIdacaqGGaGaaeOyaiaabwga caqGJbGaaeyyaiaabwhacaqGZbGaaeyzaiaabccacaaI4aWaaWbaaS qabeaacaaIYaaaaOGaeyypa0JaaGOnaiaaisdacaGGUaaaaa@4600@

But also: 64 = 4 16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI2aGaaGinaaWcbeaakiabg2da9maakaaaqqaaaaaaOpGqSvxza8qa baGaaGinaaWcpaqabaGccqGHflY1daGcaaaeeG+aaaaaaivzKbWdce aacaaIXaGaaGOnaaWcpaqabaaaaa@4243@

And this simplifies to:

64 =24=8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI2aGaaGinaaWcbeaakiabg2da9abbaaaaaG+acXwDLbWdbiaaikda paGaeyyXICneeG+aaaaaaivzKbWdciaaisdapaGaeyypa0JaaGioaa aa@430B@

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: 9 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoamaaka aabaGaaGioaaWcbeaaaaa@3797@

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?

9 8 =9 4 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoamaaka aabaGaaGioaaWcbeaakiabg2da9iaaiMdadaGcaaqaaiaaisdaaSqa baGccqGHflY1daGcaaqaaiaaikdaaSqabaaaaa@3D6E@

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).

9 4 2 92 2 18 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI5a WaaOaaaeaacaaI0aaaleqaaOGaeyyXIC9aaOaaaeaacaaIYaaaleqa aaGcbaGaaGyoaiabgwSixlaaikdacqGHflY1daGcaaqaaiaaikdaaS qabaaakeaacaaIXaGaaGioamaakaaabaGaaGOmaaWcbeaaaaaa@4418@

Example 2:

Simplify 8 32 x 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaaaaa@3C84@

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.

8 32 x 4 =8× 32 x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaGccqGH9aqpcaaI4aGaey41aq7aaSaaae aadaGcaaqaaiaaiodacaaIYaaaleqaaaGcbaWaaOaaaeaacaWG4bWa aWbaaSqabeaacaaI0aaaaaqabaaaaaaa@4412@

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

8 32 x 4 =8× 16 × 2 x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaGccqGH9aqpcaaI4aGaey41aq7aaSaaae aadaGcaaqaaiaaigdacaaI2aaaleqaaOGaey41aq7aaOaaaeaacaaI YaaaleqaaaGcbaWaaOaaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaa qabaaaaaaa@470C@

Let’s answer the square root questions we can answer:

 

8 32 x 4 =8× 4× 2 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaGccqGH9aqpcaaI4aGaey41aq7aaSaaae aacaaI0aGaey41aq7aaOaaaeaacaaIYaaaleqaaaGcbaGaamiEamaa CaaaleqabaGaaGOmaaaaaaaaaa@4618@

Notice that 8 is a fraction 8/1.

8 32 x 4 = 8 1 × 4× 2 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaGccqGH9aqpdaWcaaqaaiaaiIdaaeaaca aIXaaaaiabgEna0oaalaaabaGaaGinaiabgEna0oaakaaabaGaaGOm aaWcbeaaaOqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaa@46E3@

Multiplication of fractions is easy as π.

8 32 x 4 = 32 2 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgw SixpaakaaabaWaaSaaaeaacaaIZaGaaGOmaaqaaiaadIhadaahaaWc beqaaiaaisdaaaaaaaqabaGccqGH9aqpdaWcaaqaaiaaiodacaaIYa WaaOaaaeaacaaIYaaaleqaaaGcbaGaamiEamaaCaaaleqabaGaaGOm aaaaaaaaaa@41E3@

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.

Time for some practice problems:

 

 

1.7 Square Roots Part 1 Practice Set 1

 

1. 125 = 2. 27 = 3. 162 = 4. 75 = 5. 45 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaiOlaiaaykW7caaMc8+aaOaaaeaacaaIXaGaaGOmaiaaiwdaaSqa baGccqGH9aqpaeaacaaIYaGaaiOlaiaaykW7caaMc8+aaOaaaeaaca aIYaGaaG4naaWcbeaakiabg2da9aqaaiaaiodacaGGUaGaaGPaVlaa ykW7daGcaaqaaiaaigdacaaI2aGaaGOmaaWcbeaakiabg2da9aqaai aaisdacaGGUaGaaGPaVlaaykW7daGcaaqaaiaaiEdacaaI1aaaleqa aOGaeyypa0dabaGaaGynaiaac6cacaaMc8UaaGPaVpaakaaabaGaaG inaiaaiwdaaSqabaGccqGH9aqpaaaa@5B57@ 6. 4 x 2 = 7. 4 x 4 = 8. 4 25 = 9. 98 = 10. 48 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI2a GaaiOlaiaaykW7daGcaaqaaiaaisdacaWG4bWaaWbaaSqabeaacaaI YaaaaaqabaGccqGH9aqpaeaacaaI3aGaaiOlaiaaykW7caaMc8+aaO aaaeaacaaI0aGaamiEamaaCaaaleqabaGaaGinaaaaaeqaaOGaeyyp a0dabaGaaGioaiaac6cacaaMc8+aaOaaaeaadaWcaaqaaiaaisdaae aacaaIYaGaaGynaaaaaSqabaGccqGH9aqpaeaacaaI5aGaaiOlaiaa ykW7caaMc8+aaOaaaeaacaaI5aGaaGioaaWcbeaakiabg2da9aqaai aaigdacaaIWaGaaiOlaiaaykW7caaMc8+aaOaaaeaacaaI0aGaaGio aaWcbeaakiabg2da9aaaaa@5AA6@ 11. 4 8 = 12.4 8 = 13.3 9 = 14.2 1 4 = 15. 300 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaGymaiaac6cacaaMc8UaaGPaVpaakaaabaGaaGinaaWcbeaakiab gwSixpaakaaabaGaaGioaaWcbeaakiabg2da9aqaaiaaigdacaaIYa GaaiOlaiaaykW7caaMc8UaaGinamaakaaabaGaaGioaaWcbeaakiab g2da9aqaaiaaigdacaaIZaGaaiOlaiaaykW7caaMc8UaaG4mamaaka aabaGaaGyoaaWcbeaakiabg2da9aqaaiaaigdacaaI0aGaaiOlaiaa ykW7caaMc8UaaGOmamaakaaabaWaaSaaaeaacaaIXaaabaGaaGinaa aaaSqabaGccqGH9aqpaeaacaaIXaGaaGynaiaac6cacaaMc8UaaGPa VpaakaaabaGaaG4maiaaicdacaaIWaaaleqaaOGaeyypa0daaaa@617C@

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.7 Square Roots Part 1, Practice 2

Simplify problems 1 through 8.

1.      24 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaaGinaaWcbeaaaaa@378C@ 2. 8 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI4aGaamiEamaaCaaaleqabaGaaG4maaaaaeqaaaaa@38B0@






3. 200 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaaGimaiaaicdaaSqabaaaaa@3842@ 4. 27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaGaaG4naaWcbeaaaaa@378F@





5. 7x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI3aGaamiEaaWcbeaaaaa@37D0@ 6. 4 a 2 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaka aabaGaamyyamaaCaaaleqabaGaaGOmaaaakiaadkgaaSqabaaaaa@3990@





 

7. 3x 98 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaGcaaqaaiaaiMdacaaI4aGaamiEamaaCaaaleqabaGaaGOmaaaa aeqaaaaa@3B2C@ 8. 1 3 9 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaG4maaaacqGHflY1daGcaaqaamaalaaabaGaaGyoaaqa aiaadIhadaahaaWcbeqaaiaaikdaaaaaaaqabaaaaa@3C92@

 

 

 

 

9. Show that a 2 b= a 4 b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiaadkgacqGH9aqpdaGcaaqaaiaadggadaah aaWcbeqaaiaaisdaaaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaqaba aaaa@3D78@ 10. Why is finding perfect squares appropriate

when simplifying square roots?

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