radical expressions. are the same as square roots
TRANSCRIPT
Radical Expressions
• Square Roots & Perfect Squares• Simplifying Radicals• Multiplying Radicals• Dividing Radicals• Adding Radicals
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.
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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.
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n
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6
€
4
Cube Root
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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.
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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)
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
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€
OF
€
PS•
Perfect Square times Other Factor
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 =
= 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
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200x 3 = Simplifying 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
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=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)
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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
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
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
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
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
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.
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
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
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
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 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
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 =
€
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
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
1. Simplify to Like • Rationalize Denominator
€
's€
3 +1
3
€
3 +1
3
= 3 +1
3⋅
3
3
= 3 +3
9
= 3 +3
3