Chapter 7 – Radical Expressions and Equations Square Roots (2)2 = à There are TWO Square Roots of 4 (-2)2 = 2 is the principal, or positive, square root and -2 is the negative square root. All positive real numbers have a principal/positive square root and a negative square root. Ex: What are the square roots of 25? -25? Radicals

is pronounced “radical 25” and symbolizes the principal square root of 25. We can simplify this expression, since we know the square roots of 25:

= 5 If we want to specify the negative square root, we write - = -5 Sometimes we won’t be able to simplify a radical because the square roots are irrational numbers. For example,

= 1.41421356… so we’ll just write instead of its decimal approximation. Ex: Simplify, if possible

:

4

4

25 : 5,

. 5- 25 : ST

,- 5i ←

Notre

4 =L4

= - 4 Not Real

= z Not Real

= - 2 = 0

The parts of a radical: Bear Math “You don’t have to be faster than the bear, you just have to be faster than your friend.”

3200 A note about absolute values:

Whether you start with 3 or -3, you end up with 3. So if we have

we don’t know whether we started with a positive number or a negative number (or 0,) so we have to use absolute value to keep our final answer positive.

Ex: Simplify

cabin8

fza←raY¥n M

€Qaar!

index eradicate Inside←

060¥

C Now :D

=

75.5222hm11

32¥.ax IEEEI's 't5¥ =4Ok* , "

in=

}F÷res

= |x :xxxxxxtxxxx-x.IE#=/x5/=

1×15

= -Ixxxyy=Ex¥IH=x4yl

NotReal

Bigger Bears – Cube Roots and Fourth Roots Ex: Simplify

Multiplying Radicals As long as the index is the same, you can easily multiply two radicals:

You can also split radicals into two products:

The rule for radicals is that, if you can simplify, then you have to simplify.

c- # people

¥g=3M¥ when

3 c- omegas

=

3xx¥

÷±,××x 6÷3= 2

xE=⇒teeter2

= 4K¥x¥3fh÷4

7.3,

7.4

Ex: Simplify

Ex: Multiply and Simplify

More on Radicals and Complex Numbers

Dividing Radicals and Simplest Radical Form

¥3¥¥*kTy

=¥E¥×EI#X""mt¥#I¥ixx=x3y5z4Fxt

=

Beait be Simplified

= FEEnanxxxxxyyyyyyyye¥¥Y 427=2*12 kg

of=F÷= 2

Ex: Divide and Simplify:

Rationalizing the Denominator

= ME =4×M⇐y4

= E fEt= I Btxy= E . 6 Mxy =3 KA

= 46¥84 = FEE TEED#

= b✓ Fa '

= oib th

= ¥=£ ← Againstthe math law

← Making sure there are no

radicals in thedenominator

±1

Ex: Rationalize the Denominators

= E¥¥F¥tI3

.¥i¥*¥e÷±#4

a¥F¥¥¥÷⇒b

TE.TK.

.# ..io#.....5.5bcccc- - - -

=ta*

Complex Numbers If x2 + 1 = 0, what is x? x2 + 1 = 0 (Can’t be factored – sum of squares) x2 = -1 x = ? Let’s define a new number i, such that i2 = -1 so i = Then we have:

i

i2

i3

i4

Remember that “Complex” means it has more than one part. Complex Number: a + bi a = Real part b = Imaginary part Ex: Write as Complex Numbers 1) i 2) -i 3) 3 + i 4) - i – 42 5) 23 i

7.7

(15¥=G)fD= 1

i= imaginary

i

==c il . i = i2 . it - 1 :c - i

= iii. Eat .til 1

i5= it . i

-÷- 0¥

- li or Ot C- Di

=3 + li

= - 42 - i or . 42+(-1 ) ;or -42 - li

= 0+23 i

Id timajnory

6) 5 Ex: Simplify

=

5:

:-

za- 2) f2)= -8

=:

:+ bit Zi

= I 1+8

: A

= 4+5 i - 6+3 i= 4-6+5 ,

+3,5=2+8;

= 2 i . 5 = 10 i

= - Zi = - Zi

= Zi .5i

= lOi2=l0fD=

-10=2.2-23.43%2.3%3.c

.

. 4 - Gitbi -9÷ 4- aih¥#= 4-94 )

C- I )= 4+9=13 y W

ts¥÷n¥4÷=?¥=¥i1+3 ; =E2t9D( ltsi ) - 2- 6it9i+27e

. -

fit -

1+3i 1+3 .c - 3i - 9.cz

w - 2+3.4279Conjugate

= #= -

2+3*1+9=

-29+3=0=¥+÷i

Rational Numbers as Exponents and Radical Equations

Note: “Assume no radicands were formed by raising negative numbers to even powers” means that you don’t need to include absolute values in your answers. Eventually someone wondered if radicals could be represented by exponents, since they seem to behave so similarly, and the following reasoning was considered:

Notice that this suggests that

Following this reasoning, we can also see that:

7.

:F- 4a [ the savage]texpomnt

4a . 2

= 4'

: ex'

'

=tz

,

#-

4±

÷te =

Ex: Simplify

÷:t÷→÷Kff8¥24 =

g÷t÷= got

orient = temerityzj±3÷= zj

'

= at

⇐H⇐kx¥y÷= x÷y÷o :#

a±= at 4A

' Fx

[( abFIab )Its

= ( ab )5

Qftagpspyt[a÷bzeI

= [ at thee ] 't

= a

't b¥cE= at btu it

�8� 3449=

×¥=×"×

's =L 6W

3 )¥i¥.

Solving Radical Equations Ex: Solve and Check

1)

2)

3)

7.5 .

check :

Mizer(r×5*1=47÷x⇒¥k¥i¥ :"

6+5=111

-5 - 5check : X=2

XXI 2 :# +5

( x -532=(+5 = A +5( x -5)(x¥ to,x+E=.x*7 = 3+5

t.sn =8 Ohno!

In ×=96- 11×+18=0

( × . zxx-97=0 oil TH +5¥9= A+5×-9=4+5✓

×=2is extraneous.