Radicals

Module 14 Topic 1

What are Perfect Squares?

1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 165 • 5 = 256 • 6 = 36

49, 64, 81, 100, 121, 144, ...and so on….

Since , . 1642 164

Finding the square root of a number and squaring a number are inverse operations.

To find the square root of a number n, you must find a number whose square is n.  For example,

is 7, since 72 = 49.49

Likewise, (–7)2 = 49, so –7 is also a square root of 49.

We would write the final answer as:We would write the final answer as: 749

The symbol, , is called a radical sign. 

An expression written with a radical sign is called a radical expression. 

The expression written under the radical sign is called the radicand.

NOTE:  Every positive real number has two real number square roots. 

The number 0 has just one square root, 0 itself. 

Negative numbers do not have real number square roots.

When evaluating we choose the positive value of a called the principal root.

13169

00

RootsRNo eal 4

Evaluate 169 13Notice, since we are evaluating, we only use the positive answer.Notice, since we are evaluating, we only use the positive answer.

For any real numbers a and b, if a2 = b,

then a is a square root of b.

abthenba 2

749 4972 then

11121 121112 then

Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations.

Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations.

2

2

2 x 2 = 4

Perfect Square

The square root of 4 is ...

2

24

3 x 3 = 9

3

3

Perfect Square

The square root of 9 is ...

3

39

4 x 4 = 16

4

4

Perfect Square

4The square root of 16 is ...

416

5

5

5 x 5 = 25

Perfect Square

Can you guess what the square root of 25 is?

5The square root of 25 is ...

525

This is great, But….

Do you really want to draw blocks for a problem like…

probably not!

211

If you are given a problem like this:

Find

Are you going to have fun getting this answer by drawing 2025 blocks? Probably not!!!!!!

2025

452025

It is easier to memorize the perfect squares up to a certain point. The following should be memorized. You will see them time and time again.

x x2 x x2

0 0 10 100

1 1 11 121

2 4 12 144

3 9 13 169

4 16 14 196

5 25 15 225

6 36 16 256

7 49 20 400

8 64 25 625

9 81 50 2500

ba

To name the negative square root of a, we say

525

ba To indicate both square roots, use the plus/minus sign which indicates positive or negative.

525

7

17

4

14

3

13

2

1

xx

xx

xx

xx

nn xx1

Simplifying Radicals

• Negative numbers do not have real number square roots.

• No Real Solution

Solution Real No a

Solution Real No25

= b

This symbol represents the principal square root of a.

The principal square root of a non-negative number is its nonnegative square root.

a

525

Gizmo: Square Roots

Simplifying Radicals235998 zyx

zzyyyxxxxx 1198Divide the number under the radical.If all numbers are not prime, continue dividing.

zzyyyxxxxx 11338

Find pairs, for a square root, under the radical and pull them out.

zyxx

zzyyyxxxxx

3

11338

Multiply the items you pulled out by anything in front of the radical sign.

Multiply anything left under the radical . xyzyxx 1138

xyyzx 1124 2It is done!

Evaluate the following:

81

25.0

4

1

999

5.05.05.0

2

1

2

1

2

1

6x 333 xxx

To solve: Find all factorsPull out pairs (using one number to represent the pair. Multiply if needed)

100

5522

52

10

100

1010

10

Find all real roots:

81

25.0

4

1

999

5.05.05.0

2

1

2

1

2

1

999

5.05.05.0

2

1

2

1

2

1

981

5.025.0

2

1

4

1

• To find the roots, you will need to simplify

radial expressions in which the radicand is not a perfect square using the Product Property of Square Roots.

baab

Not all numbers are perfect squaresNot all numbers are perfect squares

THIS IS WHERE KNOWING THE PERFECT SQUARES IS VITAL

x x2 x x2

0 0 10 100

1 1 11 121

2 4 12 144

3 9 13 169

4 16 14 196

5 25 15 225

6 36 16 256

7 49 20 400

8 64 25 625

9 81 50 2500

Gizmo: Simplifying Radicals

Examples:

A.Simplify 50

Steps Explanation

25

225

25550

25Simplify

SquarePerfect A - 25(5)(5)

ionFactorizat Prime

B. Simplify 147

Steps Explanation

37

349

377147

49Simplify

SquarePerfect A - 49(7)(7)

ionFactorizat Prime

yxxy

The general rule for reducing the radicand is to remove any perfect powers.

We are only considering square roots here, so what we are looking for is any factor that is a perfect square.

In the following examples we will assume that x is positive.

Gizmo: Simplifying Radicals

Examples:

A. Evaluate

B. Evaluate

xxxx 4441616

xxxxxxx 223

. xof square theiswhich

, xoffactor a hasit square,perfect anot is Although x 23

x16

3x

xxxxxxxxxx 222445

. xof square

theis which , xisfactor squareperfect theHere2

4

xxxxxxx 2224248 2445

x.offactor one and 2 a behind leaving , xof

factor a and 4 aout takecould weexample In this2

Examples:

C. Evaluate

D. Evaluate

5x

58x

Examples:

E.

3 3780 yx3 52222 yyyxxxxxxx3 )())((52)222( yyyxxxxxxx

32 52 2 xyx 32 10 2 xyx

Unless otherwise stated, when simplifying expressions using variables, we must use absolute value signs.

aan n when n is even.

*All the sets of “3” have been grouped. They are cubes!

NOTE:  No absolute value signs are needed when finding cube roots, because a real number has just one cube root.  The cube root of a positive number is positive.  The cube root of a negative number is negative.

Evaluate the following:

16

449x

23

8

1yx

92559 m

No real roots

222 777 xxx

xxyyyxxx4

1

4

1

4

1

mmmmm 22225159

mmm 2559 22 mm 2559 4

What are Cubes?

• 13 = 1 x 1 x 1 = 1• 23 = 2 x 2 x 2 = 8• 33 = 3 x 3 x 3 = 27• 43 = 4 x 4 x 4 = 64• 53 = 5 x 5 x 5 = 125

• and so on and on and on…..

1 23 4

5 67 8

Cubes

2

2

2

2 x 2 x 2 = 8

823

3 x 3 x 3 = 27

2733

3

3

3

Nth Roots

When there is no index number, n, it is understood to be a 2 or square root.

For example: = principal square root of x.

Not every radical is a square root.

If there is an index number n other than the number 2, then you have a root other than a square root.

x

• Since 32 = 9. we call 3 the square root of 9.

• Since 33 =27 we call 3 the cube root of 27.

• Since 34 = 81, we call 3 the fourth root of 81.

Nth Roots

39

3273

3814

More Explanation of Roots

n ba

• This leads us to the definition of the nth root of a number. If an = b then a is the nth root b notated as, .

Nth Roots

• Since (-)(-) = + and (+)(+) = + , then all positive real numbers have two square roots.

• Remember in our Real Number System the is not defined.

• However we can find the cube root of negative numbers since (-)(-)(-) = a negative and (+)(+)(+) = a positive.

• Therefore, cube roots only have one root.

b

Nth Roots

Type of Number Number of Real nth Roots when

n is even

Number of Real nth Roots when

n is odd.+ 2 1

0 1 1

- None 1

Nth Roots of Variables• Lets use a table to see the pattern when

simplifying nth roots of variables.2x

4x

3 3x

3 6x

n mx

xx

xxxx 22

3 xxx

xxxx 3 33

x

2xx2xn

mx

*Note: In the first row above, the absolute value of x yields the principal root in the event that x is negative.

Examples:

A.Find all real cube roots of -125, 64, 0 and 9.

B.Find all real fourth roots of 16, 625, -1 and 0.

3 9 and 0 4, 5,- :Solutions

0 and Undefined5, 2, : Solutions

As previously stated when a number has two real roots, the positive root is called the principal root and the Radical indicates the principal root. Therefore when asked to find the nth root of a number we always choose the principal root.

F. 3 931000 yxSimplfy

3 33333 93 )(101000 yxyx Write each factor as a cube.

3 33)10( xy Write as the cube of a product.

310xy Simplify.

Absolute Value signs are NOT needed here because the index, n, is odd.

Application/Critical Thinking

A. The formula for the volume of a sphere is . Find the radius, to the nearest hundredth, of a sphere with a volume of

.

B. A student visiting the Sears Tower Skydeck is 1353 feet above the ground. Find the distance the student can see to the horizon. Use the formula to the approximate the distance d in miles to the horizon when h is the height of the viewer’s eyes above the ground in feet. Round to the nearest mile.

C. A square garden plot has an area of .a. Find the length of each side in simplest radical form.b. Calculate the length of each side to the nearest tenth of a foot.

3 3

4rV

3in 15

hd 5.1

2 24 ft

Application Solutions:

A. B.

inr

r

r

r

r

r

53.114.3

25.11

14.325.11

4

45

3

4

4

3)15(

4

3

3

415

3

3

3

3

3

milesd

d

d

hd

45

5.2029

)1353)(5.1(

5.1

C.

ftb

sa

s

s

sA

9.4 )

62 )

24

24 2

2

Evaluate the following:

4 162

2222

444

4

To solve: Find all factorsPull out set’s that contain the same number of terms as the root (using one number to represent the set of 4. Multiply if needed)

814

994

3

3 1000

1010103

10

33334

4 4xx

4 xxxx

3 64

4

22

222222

4242883

33

3 2166

6663

Evaluate the following:

2

1

144 121441442

1

2

1

)144( No real roots

516807

32

7

2

7

255

5

5 31520486 pnm

5 31520)2243( pnm

5 315205 2)3( pnm

5 334 2 3 pnm

Practice Problems and Answers

Solving EquationsWhen solving equations with exponents, you must isolate the variable (with the exponent).

Then you must take the appropriate root of both sides of the equation.

362 n

362 n

Since the square and the square root are inverse operations, they cancel each other, as can bee seen on the left side of the equation.36n

6n

To check your solutions:

Plug both answers into the original equation.

3636

36)6( 2

362 n

3636

36)6( 2

Both answers, 6 and -6, work.

0502 2 x

502 2 x

252 x

5x

71687 5 n

10245 n

55 4n

5 55 5 4n

4n

Did you check your answers by plugging both answers into the original equation.

10245 n

51

5

15 1024n

51

1 1024n

4n

Alternative Method

If you did, you found that for the equation, , -4 does NOT work!!!!!!!When you plug in -4 for n, you get – 7168, which is not what was given in the equation. So, n = 4 works, n = -4 does not.

The solution is n = 4 and the extraneous solution is n = -4.

Extraneous solutions do not satisfy original equation and must be discarded.

71687 5 n

71687 5 n

Did you check your answers by plugging both answers into the original equation.

125184 x

1434 x

46.3x

77762 5 x

62 x

4x

21879 5 d

2435 d

3d

Solve:

The following equation is used by ABC Toys to determine how many pieces of a specific round toy will fit into a shipping crate.

3 8

3rP

Find the approximate radius of each toy, rounded to the nearest hundredths, if you know that there are 50 toys in the box.

3 8

350 r

3 3400 r

3 44131816.42 r

Multiply both sides by 8 to get rid of the fraction.Multiply both sides by 8 to get rid of the fraction.

Divide both sides by 3∏Divide both sides by 3∏

r 48815912.3

r 49.3

Take the cube root of both sidesTake the cube root of both sides

Round to the correct place valueRound to the correct place value

Plug in what you knowPlug in what you know

A square piece of land is being turned into a skate park.

The area of the land piece is 189 ft2.

a) Find the length of each side in simplest radical form.

b) Calculate the length of each side to the nearest tenth of a foot.

a) Find the length of each side in simplest radical form.

The area of a square is foundby using the equation, A =s2.

Plug in what you know, and solvefor s.

b) Calculate the length of each side to the nearest tenth of a foot.

sft

sft

sA

213

189 22

2

ft7.13213

13.7 ft 13.7 ft