Ch ap t er 7 Exponents and
Exponential Functions
Zero as an Exponent
For every nonzero n u m b e r a, a 0 = 1.
Negative Exponent
For every nonzero n u m b e r a and ra tio n a l n u m b e r n,
Multiplying Powers With the Same Base
For every nonzero n u m b e r a and ra tio n a l num bers m and n,
am. an = am + n
Dividing Powers W ith the Same Base
For every nonzero n u m b e r a and ra tional num bers m and n,Raising a Pow er to a Pow er
For every nonzero n u m b e r a and ra tional num bers m and n,
(am)n = amn.
Raising a Product to a Pow er
For every nonzero n u m b e r a and b and ra tio n a l n u m b e r n,
(ab)n = anbn.
Raising a Q uo tien t to a Pow er
For every n o n zero n u m b e r a and b and rational number n,
(a)n = aZ
[bj bn'
Properties of Rational Exponents
If th e n th ro o t o f a is a real n u m b e r and m is an integer, th e n
^ = ~{/a and ^ = 1v/a™ = (’{/a)m. If m is negative,
a + 0.Exponential Growth and Decay
An exponential function has the form y = a • b*, where a is a
nonzero constant, b is greater than 0 and not equal to 1, and
x is a real num ber.
The function y = a • b*, where b is the growth factor,
models exponential growth for a > 0 and b > 1.
The function y = a • b*, where b is the decay factor,
models exponential decay for a > 0 and 0 < b < 1.Geometric Sequence
The explicit form fo r the rule o f a geom etric sequence is
A(n) = a • rn ~ 1, where A(n) is the nth term, a is the first
term , n is the term number, and r is the common ratio.
The recursive form fo r the rule o f a geom etric sequence is
A(n) = A{n - 1) • r; A(1) = a, where A(n) is the nth term, a
is th e firs t te rm , n is th e te rm num ber, and r is the common
ratio.Factoring Special Cases
For every non ze ro n u m b e r a and b:
a2 - b2 = (a + b)(a - b)
a2 + lab + b2 = (a + b)(a + b) - (a + b)2
a2 - lab + b2 = (a - b)(a - b) = (a - b)2Ch ap t er 9 Quadratic Functions
and Equations
Graph of a Quadratic Function
The graph o f y = ax2 + bx + c, w h e re a f 0, has th e line
x = as its axis o f sym m etry. The x -c o o rd in a te o f the
vertex is j 2.
Zero-Product Property
For every real n u m b e r a and b, if ab = 0, th e n
a = 0 o r b = 0.
Quadratic Formula
If ax2 + bx + c = 0, w h e re a + 0, th e n
_ -b + Vb2"- 4ac
x ~ 2 a
Property of the Discriminant
For th e q u a d ra tic e q u a tio n ax2 + bx + c = 0, w h e re a f 0,
the value o f the discrim inant b2 - 4 ac tells you th e n u m b e r
of solutions.
If b2 - 4 ac > 0, th e re are tw o real solutions.
If b2 - 4 ac = 0, th e re is one real so lu tio n.
If b2 - 4 ac < 0, th e re are no real solutions.Ch ap t er 10 Radical Expressions
and Equations
The Pythagorean Theorem
In a rig h t tria n g le , th e sum o f th e squares o f th e lengths
of the legs is equal to the square of the length of the
hypotenuse: a2 + b2 = c2.
The Converse of the Pythagorean Theorem
If a tria n g le has sides o f lengths a, b, and c, and
a2 + b2 = c2, th e n th e tria n g le is a rig h t tria n g le w ith
hypotenuse of length c.
Multiplication Property of Square Roots
For every n u m b e r a a 0 and f t > 0 , V a b = V a • V b.Chapter 8 Polynomials and Factoring