Download Lesson 2: Negative exponents, product and power, theorems for

Lesson 2:
Negative exponents, product and
power, theorems for exponents,
circle relationships
We define 2 to the third power as
follows:
3
2=2x2x2
We have agreed that 2 means 2
times 2 times 2.
3
In a similar fashion, we define 2 to
the negative third power to mean 1
over 2 to the third power.
2-3 = 1/2 3
-n
Definition of x : if n is any real
number and x is any real number
that is not zero,
-n
x = 1/x
n
This definition tells us that when we
write an exponential expression in
reciprocal form, the sign of the
exponent must be changed. If the
exponent is negative, it is positive in
reciprocal form; and if it is positive, it
is negative in reciprocal form. We
also say x cannot be zero since
division by zero is undefined.
Example:
Simplify
a)
b)
c)
d)
e)
1/3
-2
-3
3
-2
-3
(-3)-2
-3
-(-3)
Answer:
a) 9
b) 1/27
c) -1/9
d) 1/9
e) 1/27
We know that x = x  x
2
3
We also know that x = x  x  x
Using these definitions we can find
the value of the expression
2
3
5
xx=xxxxx=x
Product Theorem for Exponents:
If m and n and x are real numbers
and x ≠ 0
m
n
xx=x
m+n
Example:
Simplify
2
-5 -4 5
0
x yx y x x
Answer:
2
-3
xy
Example:
Simplify
-3 4
5 -10
yy x y x
-6 -3
10 2
yx yx
Answer:
-1 -5
yx
We can use the product theorem
2 3
2
2
2
6
to expand (x ) = x  x  x = x
Power Theorem for Exponents:
If m and n and x are real numbers
m n
mn
(x ) = x
Extension of the Power Theorem: if
m a b c are
n real
mn numbers
an bn cn
the variables
(x y z k …) = x y z k …
Example:
Simplify
-3 2
-2 -3
x(x ) y(xy )
2 3
-3
2 3
(y ) y (x )
Answer:
y
x
4
14
If we know the area of a circle, we
can find the diameter of the circle
and can find the radius of the
circle. If we know the
circumference of a circle, we can
also find the diameter and the
radius of the circle.
Example:
The area of a circle is 12.2 m.
What is the approximate
circumference of the circle?
2
Answer:
πr = Area = 12.2
2
r = 12.2/π
2
r = √(12.2/π)
r = 1.97
Circumference = 2πr ≈ 2π(1.97)
≈ 12.38 meters
Example:
The circumference of a circle is 8π
cm. What is the area of the circle?
Answer:
8π = 2πr
8π/2π = r
4 cm = r
Area = π(4cm)
= 16πcm
2
2
HW: Lesson 2 #1-30