radical expressions. are the same as square roots

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RADICAL EXPRESSIONS

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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

What is the square root?

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

X2

X

X

x 2 = x

64x2

64x 2

What is the square root?

Answer

64x2

8x

8x

64x 2

= 8x

4x2

4x 2

What is the square root?

Answer

4x2

2x

2x

4x 2 = 2x

(x+3)2

(x + 3)2 =

What is the square root?

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

Recognizing Perfect Squares(NAME THE SQUARE ROOTS)

4 x2

4x625x2

Recognizing Perfect Squares(NAME THE SQUARE ROOTS)

2

2 4

x

x x2

2x3

2x3 4x6

5x

5x 25x2

Simplifying Square Roots

Simplifying Square Roots

8 =

We can simplify if 8 contains a Perfect Square

Simplifying Square Roots

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

Simplifying Square Roots

8 =

Simplifying Square Roots

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•

Perfect Square times Other Factor

Simplifying Square Roots

8 =

= 4 2

= 2 2

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

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

2. Take square root of first radical

8 =

Simplifying Square Roots

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

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

2. Take square root of first radical

48 =

Simplifying Square Roots

48 =

= 16 3

= 4 3

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

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

2. Take square root of first radical

48 =

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Simplifying Square Roots

27x 2 =

Simplifying Square Roots

27x 2 =

= 9x 2 3

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

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

27x 2 =

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Simplifying Square Roots

27x 2 =

= 9x 2 3

= 3x 3

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

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

2. Take square root of first radical

27x 2 =

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Simplifying Square Roots

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

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

200x 3 =

Simplifying Radicals

= 100x 2 2x

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

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

200x 3 =

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

= 100x 2 2x

=10x 2x

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

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

2. Take square root of first radical

200x 3 = Simplifying Radicals

200x 3

25

Simplifying Fraction Radicals

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

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

200x 3

25

Simplifying Fraction Radicals

25 = 5

200x 3

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

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

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

2. Take square root of first radical

200x 3

25

Simplifying Fraction Radicals

25 = 5

200x 3

=10x 2x

5

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

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

2. Take square root of first radical

3. REDUCE

200x 3

25

Simplifying Fraction Radicals

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

1. Multiply to one radical

6m ⋅ 8m

6m ⋅ 8m

= 48m2

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

6m ⋅ 8m

6m ⋅ 8m

= 48m2

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

6m ⋅ 8m

6m ⋅ 8m

= 48m2

= 16m2 ⋅ 3

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

• Take square root of first radical

6m ⋅ 8m

6m ⋅ 8m

= 48m2

= 16m2 ⋅ 3

= 4m 3

Multiply Radicals

2 ⋅ 14

Multiply Radicals

1. Multiply to one radical

2 ⋅ 14

Multiply Radicals

1. Multiply to one radical

2 ⋅ 14

2 ⋅ 14

= 28

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

2 ⋅ 14

2 ⋅ 14

= 28

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

• Take square root of first radical

2 ⋅ 14

2 ⋅ 14

= 28

= 4 ⋅ 7

= 2 7Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Multiply Radicals

5 5 ⋅ 10

Multiply Radicals

1. Multiply to one radical

5 5 ⋅ 10

's

Multiply Radicals

1. Multiply to one radical

5 5 ⋅ 10

5 5 ⋅ 10

= 5 50€

's

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

5 5 ⋅ 10

5 5 ⋅ 10

= 5 50

=5 25 ⋅ 2

's

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

• Take square root of first radical

5 5 ⋅ 10

5 5 ⋅ 10

= 5 50

=5 25 ⋅ 2

= 5 ⋅5 2

= 25 2

's

Multiply Radicals

1. Multiply to one radical

18 ⋅ 80

's

Multiply Radicals

1. Multiply to one radical

18 ⋅ 80

18 ⋅ 80

= 1440 €

's

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

18 ⋅ 80

18 ⋅ 80

= 1440

= 144 ⋅ 10

's

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Multiply Radicals

1. Multiply to one radical

2. Simplify

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

• Take square root of first radical

18 ⋅ 80

18 ⋅ 80

= 1440

= 144 ⋅ 10

=12 10

's

Another Way to Multiply Radicals

Multiply Radicals By Factoring Perfect

Squares First

18 ⋅ 80

Multiply Radicals By Factoring Perfect

Squares First

1. Factor any perfect squares

18 ⋅ 80

Multiply Radicals By Factoring Perfect

Squares First

1. Factor any perfect squares

18 ⋅ 80

18 ⋅ 80

= 9 2 ⋅ 16 5

Multiply Radicals By Factoring Perfect

Squares First

1. Factor any perfect squares

2. Take square roots of perfect squares€

18 ⋅ 80

18 ⋅ 80

= 9 2 ⋅ 16 5

= 3 2 ⋅4 5

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Multiply Radicals By Factoring Perfect

Squares First

1. Factor any perfect squares

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( )

Multiply Radicals with Distributive Prop.

1. Mult. With Distr. Property

7x x + 2 7( )

Multiply Radicals with Distributive Prop.

1. Mult. With Distr. Property

7x x + 2 7( )

= 7x ⋅ x + 7x ⋅2 7

= 7x 2 + 2 49x

Multiply Radicals with Distributive Prop.

1. Mult. With Distr. Property

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•

Perfect Square times Other Factor

Multiply Radicals with Distributive Prop.

1. Mult. With Distr. Property

2. Factor any perfect squares

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

Multiply Radicals with Distributive Prop.

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

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

2. Take square root of first radical.

3. Reduce

a4

64

a4

64=

a4

64=a2

8

Dividing Square Roots

25a3

9

Dividing Square Roots

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

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

25a3

9

Dividing Square Roots

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

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

25a3

9

25a3

9=

25a2 a

9

Visual to REMEMBER

OF

PS•

Perfect Square times Other Factor

Dividing Square Roots

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

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

2. Take square root of first radical.

25a3

9

25a3

9=

25a2 a

9

=5a a

3=

Dividing Square Roots

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

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

2. Take square root of first radical.

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•

Perfect Square times Other Factor

Divide RadicalsRationalizing the

Denominator

2

3

Divide RadicalsRationalizing the

Denominator

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

2

3

Divide RadicalsRationalizing the

Denominator

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

2

3⋅

3

3

=6

9

2

3

Divide RadicalsRationalizing the

Denominator

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

2. Take the square root of the Perfect Square

3. Simplify

2

3⋅

3

3

=6

9

=6

3

2

3

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

2. Take the square root of the Perfect Square

3. Simplify

212

7

= 212

7•

7

7

= 2114

49

=21 14

7= 3 14

212

7

Divide RadicalsRationalizing the

Denominator

2

8 − 3

Divide RadicalsRationalizing the

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

Divide RadicalsRationalizing the

Denominator(2 term)

2

8 − 3⋅

8 + 3

8 + 3=

The conjugate of a binomial is SWITCHING THE SIGN BETWEEN THEM

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

2

8 − 3

2

8 − 3⋅

8 + 3

8 + 3

=16 + 2 3

64 − 9=

Divide RadicalsRationalizing the

Denominator(2 term)

8

8 64

3

− 3

8 3

−8 3

− 9 = −3

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

2. Take the square root of the Perfect Square

2

8 − 3

2

8 − 3⋅

8 + 3

8 + 3

=16 + 2 3

64 − 9

=16 + 2 3

64 − 3=

Divide RadicalsRationalizing the

Denominator(2 term)

Divide RadicalsRationalizing the

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

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

2. Take the square root of the Perfect Square

3. Simplify

Square Roots of Perfect Squares

1. Factor any 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

Values that make it Real

Values that make it Real

1. Set term in sq. root ≥ 0

2. Solve

x 2 +16 ≥ 0

x 2 ≥ −16

x is ALL REAL NUMBERS

x2 + 16

Values that make it Real

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

Simplifying Square Roots

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

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

2. Take square root of first radical

−9 =

Simplifying Negative Square Roots

−9 =

= 9 −1

= 3i

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

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

2. Take square root of first radical

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

5 8 +15 2 Adding Radicals

1. Simplify to Like

's

5 8 +15 2 Adding Radicals

=5 4 2 +15 2

= 5 ⋅2 2 +15 2

=10 2 +15 2

1. Simplify to Like

's

5 8 +15 2 Adding Radicals

=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

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

1. Simplify to Like • GCF

's

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

1. Simplify to Like • GCF

's

5 8x + 8 +15 2x + 2

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

=

=

=

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

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)

=

=

=

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

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)

=

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

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

1. Simplify to Like • Rationalize Denominator

's€

3 +1

3

3 +1

3

= 3 +1

3⋅

3

3

= 3 +3

9

= 3 +3

3

Adding Radicals

1. Simplify to Like • Rationalize Denominator

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