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CHAPTER 3
EXPONENTS
When numbers are repeatedly multiplied
together, we use exponent to illustrate this
process.
power, index, or exponent
3 x 3 x 3 x 3 x 3 = 35
base
If n is a positive integer, then an is the
product of n factors of a.
an = a x a x a x a x . . . x a
n factors
We say that a is the base, and n is the
exponent or index.
Negative Bases
A negative base raised to an odd exponent
is negative.
A negative base raised to an even
exponent is positive.
Exercise 3A
3. Simplify
a. (-1)5
i. -25
k. (-5)4
5. Use your calculator to find the values of
the following:
a. 9-1
f. 1/34
h. (0.366)0
a m  a n  a mn
am
mn

a
,a  0
n
a
m n
a 
 a mn
ab n
 a nb n
n
To multiply numbers with the same base,
keep the base and add the exponents.
To divide numbers with the same base,
keep the base and subtract the
exponents.
When raising a power to a power, keep
the base and multiply the exponents.
The power of a product is the product
of the powers.
an
a
   n ,b  0
The power of a quotient is the quotient of
b
b
the powers.
a 0  1, a  0
Any non-zero number raised to the power
of zero is 1.
1
n
a  n
a
1
1
1
n
 ,a  0
a
and in particular a
n
a
a
Exercise 3B
1. Simplify using the laws of exponents:
a. 54 x 57
k8
c. 3
k
i. (5t)3
2. Write as power of 2
a. 4
e. 32
j. 1/64
3. Write as power of 3
h. 1/81
j. 243
4. Write as a single power of 2:
d. (2x+1)2
g.
2m
m
2
5. Write as a single power of 3:
a
i.
9
1 a
3
7. Write the following in simplest form,
without brackets:
  4a
g. 
 b
3
 3p
h.  3
 q
2



2



2
8. Write without negative exponents:
 a 2b 1 
f.   2 
 c 
9. Write in non-fractional form:
a n
e. 2  n
a
Rational exponents
8.
n
1
n
a n a
a
where reads “the nth root of a”,
nєZ.
1
2
For example:
a  a
1
3
a 3 a
Rational exponents
m
n
a  a
n
m
for a > 0, n є Z+ , m є Z
Exercise 3C
1. Write as a single power of 2:
1
a. 5
2
f. 2 x
i.
3
1
3
16
2
3. Write the following in the form ax where a
is a prime number and x is rational:
g.
4
1
27
The Expansion Laws
a (b  c)  ab  ac
(a  b)(c  d )  ac  ad  bc  bd
(a  b)(a  b)  a  b
2
2
(a  b)  a  2ab  b
2
(a  b)  a  2ab  b
2
2
2
2
2
Exercise 3D.1
1. Expand and simplify:
e. 3 x (2  3 x )
1
2
3
2
1
2
f. x ( x  2 x  3 x

1
2
)
2. Expand and simplify:
b.
i.
(3x  2)(3x  5)
1
2

1
2
1
2

1
2
( x  x )( x  x )
Exercise 3D.2
1. Factorize:
a.
e.
52x  5 x
6
n 2
-6
2. Factorize:
e. 9x – 4x
f. 4x + 6(2x) + 9
i. 25x – 4(5x) + 4
3. Factorize
f. 49x – 7x+1 + 12
5. Simplify
a.
6m  2 m
2m
f.
5n 1  5n
4
6. Simplify
a.
b.
Exponential equation
An exponential equation is an equation in
which the unknown occurs as part of the
index or exponent.
For example: 2x = 8 and 30 x 3x = 7
Exponential equation
If the base numbers are the same, we can
equate exponents.
If ax = ak then x = k.
Exercise 3E
1. Solve for x
a. 2x = 8
f.
j.
2  2
x
3
x 1
1

27
2. Solve for x
e.
g.
1
27 
9
x
4
x 2
 128
1x
n.
1
 
4
 32
3. Solve for x, if possible:
a.
42 x 1  81 x
c.
2 8
x
1x
1

4
5. Solve for x:
a.
4 x  6( 2 x )  8  0
e. 25
x
 23(5 )  50  0
x
Exponential Functions
16.The most simple general exponential
function has the form
y = bx where b > 0, b ≠1.
17.
y = 2x
y = 2x is ‘asymptotic to the x-axis’ or
‘y = 0 is a horizontal asymptote’.
18. For the general exponential function
y = a • bx-c + d where b > 0, b ≠ 1, a ≠ 0:
b controls how steeply the graph increases or
decreases
c controls horizontal translation
d controls vertical translation
the equation of the horizontal asymptote is y = d
if a > 0, b > 1
the function is
increasing.
if a < 0, b > 1
the function is
decreasing.
if a > 0, 0 < b < 1
the function is
decreasing.
if a < 0, 0 < b < 1
the function is
increasing.
Exercise 3F
1. Use the graph above to
estimate the value of:
a. 21/2 or √2
c. 21.5
6. For each of the functions below:
i. sketch the graph of the function
ii. state the domain and range
iii. use your calculator to find the value of y
when x = √2
iv. discuss the behavior of y as x∞
a. y = 2x + 1
b. y = 2 – 2x
Growth and Decay
20. Exponential functions are often used to
model growth and decay.
Populations of animals, people, and bacteria
usually grow in an exponential way.
Radioactive substances, and items that
depreciate in value, usually decay
exponentially.
Growth
21. Consider a population of 100 mice
which under favorable conditions is
increasing by 20% each week.
Determine the exponential function for this
situation.
Week
Population
0
P0 = 100
1
P1 = P0 x 1.2 = 100 x 1.2
2
P2 = P1 x 1.2 = 100 x (1.2)2
3
P3 = P2 x 1.2 = 100 x (1.2)3
4
P4 = P3 x 1.2 = 100 x (1.2)4 and so on.
Pn = 100 x (1.2)n
Exercise 3G.1
1. The weight Wt of bacteria in a culture t hours after
establishment is given by Wt = 100 x 20.1t grams.
a. Find the initial weight.
b. Find the weight after:
i. 4 hours ii. 10 hours iii. 24 hours.
c. Sketch the graph of Wt against t using the
results of a and b only.
d. Use technology to graph Y1 = 100 x 20.1x and
check your answers to a, b, and c.
3. A species of bear is introduced to a large island off
Alaska where previously there were no bears. 6 pairs of
bears were introduced in 1998. It is expected that the
population will increase according to Bt = B0 x 20.18t where t
is the time since the introduction.
a. Find B0.
b. Find the expected bear population in 2018.
c. Find the expected percentage increase from
2008 to 2018.
23. Consider a radioactive substance with
original weight 20 grams. It decays or
reduces by 5% each year. The multiplier for
this is 95% or 0.95.
Year
Weight (grams)
0
W0 = 20
1
W1 = W0 x 0.95 = 20 x 0.95
2
W2 = W1 x 0.95 = 20 x (0.95)2
3
W3 = W2 x 0.95 = 20 x (0.95)3
4
W4 = W3 x 0.95 = 20 x (0.95)4 and so on.
Wn = 20 x (0.95)n
Exercise 3G.2
1. The weight of a radioactive substance t years after
being set aside is given by W(t) = 250 x (0.998)t grams.
a How much radioactive substance was initially set
aside?
b. Determine the weight of the substance after:
i. 400 years ii. 800 years iii. 1200 years.
c. Sketch the graph of W(t) for t > 0 using a and b only.
d. Use your graph or graphics calculator to find how long
it takes for the substance to decay to 125 grams.
The Natural Exponential ex
25. What is e?
It is a mathematical constant that has a
value of
2.71828182845904523536028747135266249775724709369995
It is called the Euler Number after a Swiss
mathematician named Leonhard Euler.
Exercise 3H
3. For the general exponential function y =
aekx, what is the y-intercept?
4. Consider y = 2ex.
a Explain why y can never be < 0.
b Find y if:
i. x = -20
ii. x = 20.
9. On the same set of axes, sketch and
clearly label the graphs of:
f : x  ex, g : x  ex-2,
h : x ex+3
State the domain and range of each
function.
12. The weight of bacteria in a culture is
given by W(t) = 2et/2 grams where t is the
time in hours after the culture was set to
grow.
a Find the weight of the culture when:
i. t = 0
ii. t = 30 min
iii. t = 1½ hours
iv. t = 6 hours.
b Use a to sketch the graph of W(t) = 2et/2 .
14. The current flowing in an electrical circuit t
seconds after it is switched off is given by
I(t) = 75e-0.15t
a. What current is still flowing in the circuit after:
i. 1 second ii. 10 seconds?
b Use your graphics calculator to sketch
I(t) = 75e-0.15t and I = 1.
c Hence find how long it will take for the current
to fall to 1 amp.