1 lesson 1 4
TRANSCRIPT
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Chapter 1RADICAL FUNCTIONS
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Math Box• Suppose a and b are real numbers and n is a positive
integer not equal to 1 such that an = b then, a is the nth root of b.
Example: 25 = 32 2 is the 5th root of 32 33 = 27 3 is the cube root of 27
52 = 25 5 is the square root of 25
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Rational Exponents: Its Roots
• If n is a POSITIVE ODD INTEGER and b is a REAL NUMBER, then b has exactly ONE REAL ROOT called principal nth root of b.
• If n is a POSITIVE EVEN INTEGER and b is a REAL NUMBER, then b has TWO NTH ROOTS (negative and positive)
• If n is EVEN POSITIVE INTEGER and b is a NEGATIVE NUMBER then b has NO REAL NTH ROOT.
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Rational Exponent: Its Definition
bbb mnn
m
nm 11
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Lets Apply the Definition!
832
1645 x27 3 3
2
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Simplify the following Rational Expressions:
423
251 2
3
923
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Activity 1:
Simplify the following Rational Exponents
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RADICALS
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For any real number a and b and all integers n>0
abn n is the index or orderb is the radicand√ is the radical sign
is the radical expressiona is the nth root of b
abn
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Radical Expression
sRadicand Index
3 4x
5 35xyx8 5
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Writing Rational Exponents form into Radical form
Rational Exponent Radical Form
b Base Radicand
n Denominator of the rational exponent Index or order
m Numerator of the rational exponent
Power of the whole radicand
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Rewrite the following Rational Exponents toRadical Form
421
x7 21
732 x3 2 3
1
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Activity B:
Rewrite the following Rational Exponents toRadical Form
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Rewrite the following Radical Form toRational Exponents
3 5 35 x
4 2x xy4
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Activity C:
Rewrite the following Radical Form to
Rational Exponents
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LAWS OF RADICALS
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Laws of Radicals1. When b ≠ 0 and n>1.
Example:
2. When b < 0 and n is even.Example:
3.Example:
bn nb
bn nb
bn bn
3 32 5 54
5 2 442
3 53 5 2
5
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Laws of Radicals4. Example:
5. Example:
6.Example:
nnn baab
n
nn
ba
ba
mnn m bb
3 8x 125
365
3
83
3 5 3 4 2
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Answer the following by applying theLaw of Radicals
1. 5.
2. 6.
3. 7.
4. 8.
3 53
6 26
5012
43
3
278
16
5 75
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Simplification of Radicals
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A radical expression is said to be simplified or in simplest form if:
• Case 1: The radicand has no factors whose indicated roots can still be taken.
• Case 2: The radicand does not contain a fraction.• Case 3: The denominator does not contain a radical
expression.• Case 4: The index or the order of the radical is in its
lowest form.
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Case 1: The radicand has no factors whose indicated roots can still be taken.
yx45
3
4
16
121.2.3.
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Case 2: The radicand does not contain a fraction
yx3
3
54411
.2.3.
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Case 3: The denominator does not contain a radical expression
1.
2.
3.3 275523
x
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Case 4: The index or the order of the radical is in its lowest form
1.
2.
3.
4. 1248
6 333
6
4
16
869
pn
zyxx
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Before Class Activity
In your Math BookPage 7
Items 1-10
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Operations of Radicals
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Similar Radicals are radicals with the same indices and radicand when simplified.
Examples:
37,234,
532
,22,2,27
3,5,2333 xxxx
xxxx
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Multiplication of Radicals
Multiplication of Radicals with the SAME INDICES.1. Multiply their radicands2. Multiply their numerical coefficients3. Retain the common indices4. Simplify the product
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Multiplication of RadicalsExamples: 1.
2.
3.
4. 132132
634
4432
35
2 3
xx x
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Its Your Turn!Warm-Up Practice
Activity APage 20
ODD Items Only
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Multiplication of Radicals
Multiplication of Radicals with the DIFFERENT INDICES.1. Make their indices the same by transforming
them to a fractional exponent.2. Take the LCD of their fractional exponents.3. Transform the radical form.
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Multiplication of RadicalsExamples: 1.
2. xx 22
233
3
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Addition and Subtraction ofSimilar Radicals
Similar Radicals are radicals with the same indices and radicand when simplified.
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Make each pair of radical SIMILAR
75,27
63,28
18,2
45,5
12,31.6.
2.7.
3.8.
4.9.
5. 10.
75,45
36,24
32,2
50,2
12,48
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Addition and Subtraction ofSimilar Radicals
Examples: 1.
2.
3.
4.
5. 313
505823
352
252724
525453
333
xxx
xxx
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Its Your Turn!Warm-Up Practice
Activity APage 13
Items 1-7
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Divisions of Radicals
Quotient Rule:
n
n
nyx
yx
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Divisions of RadicalsSimplify: 1.
2.251593
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Divisions of RadicalsSimplify: 1.
2.2712188
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Divisions of RadicalsSimplify: 1.
2.
3.250
832
630
2
3
xx
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Rationalizing the Denominator
Rationalize: 1.
2.
3.
4. 23
1052632
55
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Its Your Turn!Warm-Up Practice
Activity BPage 27
Items 1-8
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Conjugate of a Denominator
• If is the denominator, the conjugate is .• If is the denominator, the conjugate is .
ba
ba
ba ba
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Give the conjugate of each expression:
1.
2.
3. 12
35
13
3
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Conjugate the denominator then multiply
1.
2.
3.123
3255213
3
x
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Its Your Turn!Warm-Up Practice
Activity CPage 27
Items 1-6