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Ch 8 - Rational & Radical Functions
Simplifying Radical Expressions
Product Property of Radicals:For any real numbers a and b,
then
baab
Simplify.
330. aA
75454. zyxB
32 4010. xyyxC
22 612. mnnmD
Quotient Property of RadicalsFor any real numbers a and b,
and b ≠ 0,
if all roots are defined.
To rationalize the denominator, you must multiply the numerator and denominator by a quantity so that the radicand has an exact root.
b
a
b
a
Simplify.
4
5.A
22
820.B
32
6.C
32
6.
knD
Two radical expressions are called like radical expressions if the radicands are alike.
dcba Conjugates are binomials in the form
and
where a, b, c, and d are rational numbers. The product of conjugates is always a rational number.
dcba
Simplify.
26723210. A
15024365. B
72547254. C
56
521.
D
nth roots
The nth root of a real number a can be written as the radical expression , where n is the index of the radical and a is the radicand.
n a
Numbers and Types of Real Roots
Case Roots Example
Odd Index 1 real root The real 3rd root of 8 is 2.The real 3rd root of -8 is -2.
Even Index, Positive Radicand
2 real roots The real 4th roots of 16 are ±2.
Even Index,Negative Radicand
0 real roots -16 has no real 4th roots.
Radicand of 0 1 root of 0 The 3rd root of 0 is 0.
Find all real roots.A.Sixth roots of 64
B.Cube roots of -216
C.Fourth roots of -1024
Properties of nth Roots
Product Property of Roots
nnn baab
Quotient Property of Roots n
n
n
b
a
b
a
Simplify each expression. Assume that all variables are positive.4 1281. xA 4
8
5
16.
xB
A rational exponent is an exponent that can be expressed as m/n, where m and n are integers and n ≠ 0.
The exponent 1/n indicates the nth root.
nn aa 1
n mmnn
m
aaa The exponent m/n
indicates the nth root raised to the mth power.
Write each expression by using rational exponents.
8 413.A5 153.B
32
125. C 53
32. D
Write each expression in radical form and simplify.
Properties of Rational Exponents
Product of Powers Property
Quotient of Powers Property
Power of a Power Property
Power of a Product Property
Power of a Quotient Property
nmn
m
aa
a nmnm aaa
m
mm
b
a
b
a
mmm baab mnnm aa
Simplify each expression.
9
11
9
7
77. A4
5
4
3
16
16.B
