RADICAL EXPRESSIONS. ARE THE SAME AS SQUARE ROOTS.

RADICAL EXPRESSIONS

RADICAL EXPRESSIONS

€

ARE THE SAME AS SQUARE ROOTS

Radical Expressions

• Square Roots & Perfect Squares• Simplifying Radicals• Multiplying Radicals• Dividing Radicals• Adding Radicals

What is the square root?

€

64

64

The square root of a number is a # when multiplied by itself equals the number

64

8

8

€

64 = 8

X2

€

x 2


A Square Root is a term when squared is the Perfect Square

X2

X

X

€

x 2 = x

64x2

€

64x 2


Answer

64x2

8x

8x

€

64x 2

= 8x

4x2

€

4x 2


Answer

4x2

2x

2x

€

4x 2 = 2x

(x+3)2

€

(x + 3)2 =


Answer

(x+3)2

(x+3)

(x+3)

€

(x + 3)2 = x + 3

Radical Sign

€

• This is the symbol for square root

• If the number 3 is on top of the v, it is called the 3rd root.

€

3

Radical Sign

• This is the 4th root.

• This is the 6th root

• If n is on top of the v, it is called the nth root.

€

n

€

6

€

4

Cube Root

€

3

If the sides of a cube are 3 inches Then the volume of the cube is 3 times 3 times 3 or 33 which is 27 cubic inches.

€

273 = 3

Numbers that are perfect squares are:

12=1, 22= 4, 32= 9, 42= 16, 52= 25, 62= 36, 72= 49, 82= 64, 92= 81, 102= 100, …

NUMBER Perfect Squares

Variables that are perfect squares are:

x2, a4, y22, x100…(Any even powered variable is a perfect square)

x50

EVEN POWEREDEXPONENTS ARESQUARES

x25

x25

Recognizing Perfect Squares(NAME THE SQUARE ROOTS)

9x50

NAME THE SQUARE ROOT

9x50

EVEN POWEREDEXPONENTS ARESQUARES

3x25

3x25


4 x2

4x625x2


2

2 4

x

x x2

2x3

2x3 4x6

5x

5x 25x2

Simplifying Square Roots


€

8 =

We can simplify if 8 contains a Perfect Square


€

8 =

= 4 2

= 2 2

Look to factor perfect squares (4, 9, 16, 25, 36…)

Put perfect squares in first radical and the other factor in 2nd.

Take square root of first radical

Visual to REMEMBER

€

€

OF

€

PS•

Perfect Square times Other Factor


€

8 =


€

8 =

= 4 2

1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)

(Put perfect squares in first radical and other factor in 2nd)

€

8 =

Visual to REMEMBER

€

€

OF

€

PS•



€

8 =

= 4 2

= 2 2


(& Put perfect squares in first radical and other factor in 2nd)

2. Take square root of first radical

€

8 =





€

48 =


€

48 =

= 16 3

= 4 3




€

48 =

Visual to REMEMBER

€

€

OF

€

PS•



€

27x 2 =


€

27x 2 =

= 9x 2 3

1. Factor any perfect squares (4, 9, 16, 25, 36…)


€

27x 2 =

Visual to REMEMBER

€

€

OF

€

PS•



€

27x 2 =

= 9x 2 3

= 3x 3




€

27x 2 =

Visual to REMEMBER

€

€

OF

€

PS•





€

200x 3 =

Simplifying Radicals

€

= 100x 2 2x



€

200x 3 =

Visual to REMEMBER

€

€

OF

€

PS•


€

= 100x 2 2x

=10x 2x




€

200x 3 = Simplifying Radicals

€

200x 3

25

Simplifying Fraction Radicals

1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM.


€

200x 3

25


€

25 = 5

€

200x 3

Visual to REMEMBER

€

€

OF

€

PS•





€

200x 3

25


€

25 = 5

€

200x 3

€

=10x 2x

5




3. REDUCE

€

200x 3

25


€

25 = 5

€

200x 3

€

=10x 2x

5

= 2x 2x

Finding Perfect Squares

The most common perfect squares are 4 & 9. USE DIVISIBILITY RULES:

A number is divisible by 4 if the last 2 digits are divisible by 4.( 23,732 is divisible by 4 since 32 is.) A number is divisible by 9 if the sum of it’s digits are.(4653 is divisible by 9 since the sum of it’s digits are.)

Finding Perfect Squares

MORE EASY TO FIND PERFECT SQUARES:

A number with an even number of Zeros.( 31,300 is divisible by 10, 70,000 is divisible by 100) A number ending in 25, 50 or 75 is divisible by 25.(425 & 350 & 775 are all divisible by 25)

Perfect Squares Guide Divisibility Rules for Perfect Squares

4: If Last 2 Digits are Divisible by 49: If Sum of Digits are Divisible by 916: Use 4 Rule25: If # ends in 25, 50, 75 or 0036: Use 4 or 9 Rule49: Look for multiple of 50 and subtract multiple64: Use 4 Rule81: Use 9 Rule100: If # ends in 00121: No Rule, Divide # by 121144: Use 4 or 9 Rule

Name_____________________________________________ Per____

SIMPLIFYING RADICALS Box answer on this sheet

1) 2) 3)

4) 5) 6) 7)

8) 9) 10) 11)

12) 13) 14)

€

3x2

€

125a2

€

36n3

€

216

€

10

€

p17

€

30

€

x4

€

20

€

−3250

Simplifying SquareRoots

€

27x2 == 9x2 3=3x 3

1. Factor any perfect squares(4, 9, 16, 25, 36…)(& Put perfect squares in firstradical and other factor in 2nd)2. Take square root of firstradical€

27x2 =

€

448

€

8a5

€

2m4m3 Visual to REMEMBER€

€

OF

€

PS•Perfect Square times Other Factor

€

25x6

Multiply Radicals

€

6m ⋅ 8m

Multiply Radicals

1. Multiply to one radical

€

6m ⋅ 8m

Multiply Radicals


€

6m ⋅ 8m

€

6m ⋅ 8m

= 48m2

Multiply Radicals


2. Simplify

• Factor any perfect squares (4, 9, 16, x2, y6…)

€

6m ⋅ 8m

€

6m ⋅ 8m

= 48m2

Multiply Radicals


2. Simplify


€

6m ⋅ 8m

€

6m ⋅ 8m

= 48m2

€

= 16m2 ⋅ 3

Visual to REMEMBER

€

€

OF

€

PS•


Multiply Radicals


2. Simplify


• Take square root of first radical

€

6m ⋅ 8m

€

6m ⋅ 8m

= 48m2

€

= 16m2 ⋅ 3

= 4m 3

Multiply Radicals

€

2 ⋅ 14

Multiply Radicals


€

2 ⋅ 14

Multiply Radicals


€

2 ⋅ 14

€

2 ⋅ 14

= 28

Multiply Radicals


2. Simplify


€

2 ⋅ 14

€

2 ⋅ 14

= 28

Multiply Radicals


2. Simplify



€

2 ⋅ 14

€

2 ⋅ 14

= 28

€

= 4 ⋅ 7

= 2 7Visual to REMEMBER

€

€

OF

€

PS•


Multiply Radicals

€

5 5 ⋅ 10

Multiply Radicals


€

5 5 ⋅ 10

€

's

Multiply Radicals


€

5 5 ⋅ 10

€

5 5 ⋅ 10

= 5 50€

's

Multiply Radicals


2. Simplify


€

5 5 ⋅ 10

€

5 5 ⋅ 10

= 5 50

€

=5 25 ⋅ 2

€

's

Visual to REMEMBER

€

€

OF

€

PS•


Multiply Radicals


2. Simplify



€

5 5 ⋅ 10

€

5 5 ⋅ 10

= 5 50

€

=5 25 ⋅ 2

= 5 ⋅5 2

= 25 2

€

's

Multiply Radicals


€

18 ⋅ 80

€

's

Multiply Radicals


€

18 ⋅ 80

€

18 ⋅ 80

= 1440 €

's

Multiply Radicals


2. Simplify


€

18 ⋅ 80

€

18 ⋅ 80

= 1440

= 144 ⋅ 10

€

's

Visual to REMEMBER

€

€

OF

€

PS•


Multiply Radicals


2. Simplify



€

18 ⋅ 80

€

18 ⋅ 80

= 1440

= 144 ⋅ 10

=12 10

€

's

Another Way to Multiply Radicals

Multiply Radicals By Factoring Perfect

Squares First

€

18 ⋅ 80


Squares First

1. Factor any perfect squares

€

18 ⋅ 80


Squares First


€

18 ⋅ 80

€

18 ⋅ 80

= 9 2 ⋅ 16 5


Squares First


2. Take square roots of perfect squares€

18 ⋅ 80

€

18 ⋅ 80

= 9 2 ⋅ 16 5

= 3 2 ⋅4 5

Visual to REMEMBER

€

€

OF

€

PS•



Squares First


2. Take square roots of perfect squares

3. Multiply #,s &

€

18 ⋅ 80

€

18 ⋅ 80

= 9 2 ⋅ 16 5

= 3 2 ⋅4 5

=12 10€

's

Multiply Radicals with Distributive Prop.

€

7x x + 2 7( )


1. Mult. With Distr. Property

€

7x x + 2 7( )



€

7x x + 2 7( )

€

= 7x ⋅ x + 7x ⋅2 7

= 7x 2 + 2 49x



2. Factor any perfect squares€

7x x + 2 7( )

€

= 7x ⋅ x + 7x ⋅2 7

= 7x 2 + 2 49x

= x 2 7 + 2 49 x

Visual to REMEMBER

€

€

OF

€

PS•





3. Take square roots of perfect squares

4. Simplify if possible

€

7x x + 2 7( )

€

= 7x ⋅ x + 7x2 7

= 7x 2 + 2 49x

= x 2 7 + 2 49 x

= x 7 + 2 ⋅7 x

= x 7 +14 x


1. Multiply Terms (Use Multiplying Boxes & CLT)

2. Simplify if possible €

3x + 4 ⋅ x − 5

€

= (3x + 4)(x − 5)

= 3x 2 −11x − 20

x

3x 3x2

€

−5

€

+4

€

−15x

€

+4x

€

−20

€

3x 2 −11x − 20

Dividing Square Roots

1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator


2. Take square root of first radical.

3. Reduce

€

a4

64

€

a4

64=

a4

64=a2

8


€

25a3

9




€

25a3

9




€

25a3

9

€

25a3

9=

25a2 a

9

Visual to REMEMBER

€

€

OF

€

PS•






€

25a3

9

€

25a3

9=

25a2 a

9

=5a a

3=





3. Reduce

€

25a3

50

€

25a3

25=

25a2 a

25

=5a a

5= a a

Divide Radicals

€

2

3

Divide Radicals

1. Divide

€

2

3

Divide Radicals

1. Divide

€

27

3=

27

3

= 9€

27

3

Divide Radicals

1. Divide

2. Simplify

€

27

3=

27

3

= 9 = 3€

27

3

Divide Radicals

1. Divide

2. Simplify

€

30x 4

6x 2=

30x 4

6x 2

= 5x 2

= x 2 5

= x 5

€

30x 4

6x 2

Visual to REMEMBER

€

€

OF

€

PS•


Divide RadicalsRationalizing the

Denominator

€

2

3


Denominator

1. Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator€

2

3


Denominator

1. Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator

€

2

3⋅

3

3

=6

9

€

2

3


Denominator


2. Take the square root of the Perfect Square

3. Simplify

€

2

3⋅

3

3

=6

9

=6

3

€

2

3



3. Simplify

€

212

7

= 212

7•

7

7

= 2114

49

=21 14

7= 3 14

€

212

7


Denominator

€

2

8 − 3


Denominator(2 term)

Rationalizing the denominator means making the denominator rational or REMOVING IRRATIONAL TERMS

1. Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator

€

2

8 − 3


Denominator(2 term)

€

2

8 − 3⋅

8 + 3

8 + 3=

The conjugate of a binomial is SWITCHING THE SIGN BETWEEN THEM


€

2

8 − 3

€

2

8 − 3⋅

8 + 3

8 + 3

=16 + 2 3

64 − 9=


Denominator(2 term)

8

8 64

€

3

€

− 3

€

8 3

€

−8 3

€

− 9 = −3



€

2

8 − 3

€

2

8 − 3⋅

8 + 3

8 + 3

=16 + 2 3

64 − 9

=16 + 2 3

64 − 3=


Denominator(2 term)


Denominator(2 term)

€

2

8 − 3

€

2

8 − 3⋅

8 + 3

8 + 3

=16 + 2 3

64 − 9

=16 + 2 3

64 − 3=

16 + 2 3

61



3. Simplify

Square Roots of Perfect Squares


2. Take square root of perfect square.

€

x 2 + 6x + 9

€

(x + 3)2

= x + 3

9

633

€

( x + 3 )( x + 3 ) = ( x + 3 )2

Values that make it Real

1. Set term in sq. root ≥ 0

2. Solve

€

x + 4

€

x + 4 ≥ 0

x ≥ −4



1. Set term in sq. root ≥ 0

2. Solve

€

x 2 +16 ≥ 0

x 2 ≥ −16

x is ALL REAL NUMBERS

€

x2 + 16


Just note that if x is any real number (all numbers on the number line)

x2 can never be negative, so

(x2+16) will always be positive.

The answer is any real number

€

x2 + 16

Negative Square Roots

€

− 9 = −3

−9 = ? Can you take the square root of a negative number?

Can you take the square root of a negative number?

Because a negative number squared is positive





€

−9 =

Simplifying Negative Square Roots

€

−9 =

= 9 −1

= 3i




3. The square root of -1 is called IMAGINARY

(We will review this later)

€

−9 =

Adding Radicals

€

x + x = ?

Adding Radicals

€

1+1= 2

€

x + x = 2 x

€

x + x = 2x

€

x + x = ?

€

x + x = 2 x

€

5 + 5 = ?

€

5 + 5 = 2 5

€

6 8 + 3 8 = ?

€

6 8 + 3 8 = 9 8

€

4 8 − 3 8 = ?

€

4 8 − 3 8 = 8

€

x + 8

€

x + 8 = x + 8

€

2 x + 8 − 3 x = ?

€

2 x + 8 − 3 x

= 8 − x

€

3 3x + 8 3x = ?

−3 y − 7 y = ?

10 11 + 88 11 = ?

€

3 3x + 8 3x =11 3x

−3 y − 7 y = −10 y

10 11 + 88 11 = 98 11

€

−12 4y − 8 4y = ?

y − 34 y = ?

−45 11 + 5 11 +10 11 = ?

€

−12 4y − 8 4y = −20 4y

y − 34 y = −33 y

−45 11 + 5 11 +10 11 = −30 11

€

5 8 +15 2 Adding Radicals

€


1. Simplify to Like

€

's

€


€

=5 4 2 +15 2

= 5 ⋅2 2 +15 2

=10 2 +15 2

1. Simplify to Like

€

's

€


€

=5 4 2 +15 2

= 5 ⋅2 2 +15 2

=10 2 +15 2

= 25 2

1. Simplify to Like

2. Add the Like

€

's

€

's

€

5 8x + 8 +15 2x + 2 Adding Radicals

€


1. Simplify to Like • GCF

€

's

€


1. Simplify to Like • GCF

€

's

€

5 8x + 8 +15 2x + 2

= 5 8(x +1) +15 2(x +1)

=

=

=

€


1. Simplify to Like • GCF •

€

PS ⋅ OF

€

's

€

5 8x + 8 +15 2x + 2

= 5 8(x +1) +15 2(x +1)

= 5 4 2(x +1) +15 2(x +1)

=

=

=

€


1. Simplify to Like • GCF • •Simplify

€

PS ⋅ OF

€

's

€

5 8x + 8 +15 2x + 2

= 5 8(x +1) +15 2(x +1)

= 5 4 2(x +1) +15 2(x +1)

= 5 ⋅2 2(x +1) +15 2(x +1)

=10 2(x +1) +15 2(x +1)

=

€


1. Simplify to Like • GCF • •Simplify

2. Add the Like Terms

€

PS ⋅ OF

€

's

€

5 8x + 8 +15 2x + 2

= 5 8(x +1) +15 2(x +1)

= 5 4 2(x +1) +15 2(x +1)

= 5 ⋅2 2(x +1) +15 2(x +1)

=10 2(x +1) +15 2(x +1)

= 25 2(x +1)

Adding Radicals

€

3 +1

3

Adding Radicals

1. Simplify to Like • Rationalize Denominator

€

's€

3 +1

3

Adding Radicals


€

's€

3 +1

3

€

3 +1

3

= 3 +1

3⋅

3

3

= 3 +3

9

= 3 +3

3

Adding Radicals


2. Add the Like Terms

€

's€

3 +1

3

€

3 +1

3

= 3 +1

3⋅

3

3

= 3 +3

9

= 3 +3

3

=3 3

3+

3

3=

4 2

3

RADICAL EXPRESSIONS. ARE THE SAME AS SQUARE ROOTS.

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