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Chapter 1RADICAL FUNCTIONS
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
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.
Rational Exponent: Its Definition
bbb mnn
m
nm 11
Lets Apply the Definition!
832
1645 x27 3 3
2
Simplify the following Rational Expressions:
423
251 2
3
923
Activity 1:
Simplify the following Rational Exponents
RADICALS
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
Radical Expression
sRadicand Index
3 4x
5 35xyx8 5
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
Rewrite the following Rational Exponents toRadical Form
421
x7 21
732 x3 2 3
1
Activity B:
Rewrite the following Rational Exponents toRadical Form
Rewrite the following Radical Form toRational Exponents
3 5 35 x
4 2x xy4
Activity C:
Rewrite the following Radical Form to
Rational Exponents
LAWS OF RADICALS
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
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
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
Simplification of Radicals
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.
Case 1: The radicand has no factors whose indicated roots can still be taken.
yx45
3
4
16
121.2.3.
Case 2: The radicand does not contain a fraction
yx3
3
54411
.2.3.
Case 3: The denominator does not contain a radical expression
1.
2.
3.3 275523
x
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
Before Class Activity
In your Math BookPage 7
Items 1-10
Operations of Radicals
Similar Radicals are radicals with the same indices and radicand when simplified.
Examples:
37,234,
532
,22,2,27
3,5,2333 xxxx
xxxx
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
Multiplication of RadicalsExamples: 1.
2.
3.
4. 132132
634
4432
35
2 3
xx x
Its Your Turn!Warm-Up Practice
Activity APage 20
ODD Items Only
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.
Multiplication of RadicalsExamples: 1.
2. xx 22
233
3
Addition and Subtraction ofSimilar Radicals
Similar Radicals are radicals with the same indices and radicand when simplified.
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
Addition and Subtraction ofSimilar Radicals
Examples: 1.
2.
3.
4.
5. 313
505823
352
252724
525453
333
xxx
xxx
Its Your Turn!Warm-Up Practice
Activity APage 13
Items 1-7
Divisions of Radicals
Quotient Rule:
n
n
nyx
yx
Divisions of RadicalsSimplify: 1.
2.251593
Divisions of RadicalsSimplify: 1.
2.2712188
Divisions of RadicalsSimplify: 1.
2.
3.250
832
630
2
3
xx
Rationalizing the Denominator
Rationalize: 1.
2.
3.
4. 23
1052632
55
Its Your Turn!Warm-Up Practice
Activity BPage 27
Items 1-8
Conjugate of a Denominator
• If is the denominator, the conjugate is .• If is the denominator, the conjugate is .
ba
ba
ba ba
Give the conjugate of each expression:
1.
2.
3. 12
35
13
3
Conjugate the denominator then multiply
1.
2.
3.123
3255213
3
x
Its Your Turn!Warm-Up Practice
Activity CPage 27
Items 1-6