Download Exponents and Radicals

Exponents and Radicals
Learning Objectives
At the end of this topic, YOU should be able to:

Define exponents.

Apply the rules of exponents and simplify the exponential functions.

Apply the rules of exponents and simplify the composite functions.

Define radicals [surds].

Apply the rules of radicals for the functions having root to add, subtract,
multiply and divide.

Simplify the radicals [surds].

Rationalize the denominator having square roots and then simplify the surds.
Unit 2: Exponents and Radicals
Exponents are a very important part of Algebra.
Definition of Exponent
Exponents are shorthand for repeated multiplication of the same thing by itself.
If n is a positive integer, an represents the product of factors, each of which is a
Examples:
x3 = x  x  x
26 = 2 x 2 x 2 x 2 x 2 x 2 = 64
General Laws of Exponents
If p and q are real numbers, the following laws hold
1. Product Rule: ap. aq = ap+q
Note: Base must be identical for the rule 1and 2
p
a
 apq
2. Quotient Rule:
q
a
3. Power of a power Rule: (ap)q = apq
4. Power of a product Rule: (ab)p = apbp
p
ap
a
5. Power of a Quotient Rule:    p
b
b
1
6. Negative Exponent Rule : a n  n
a
7. Zero Exponent Rule: a0 = 1
Examples of rules
Product Rule
The exponent "product rule" tells us that, when multiplying two powers that have the same base,
you can add the exponents. In this example, you can see how it works. Adding the exponents is
just a short cut!
ap. aq = ap+q
4243 = 4.4.4.4.4
42+3 = 45
Quotient Rule
The quotient rule tells us that we can divide two powers with the same base by subtracting the
exponents. You can see why this works if you study the example shown.
ap
a
q
 apq
Power of a power Rule
The "power rule" tells us that to raise a power to a power, just multiply the exponents. Here you
see that 52 raised to the 3rd power is equal to 56.
(ap)q = apq
(52)3 = 52x3 = 56
Power of a product Rule
When a product has an exponent, each factor is raised to that power.
(ab)p = apbp
(mn)5 = m5 n5
Power of a Quotient Rule
When a Quotient has an exponent, each factor is raised to that power.
p
ap
a
   p
b
b
5
25
32
2


 
5
243
3
3
Negative Exponent Rule
Any nonzero number raised to a negative power equals its reciprocal raised to the opposite
positive power.
1
: a n  n
a
Zero Exponent Rule
According to the "zero rule," any nonzero number raised to the power of zero equals 1.
a0 = 1
Review questions: Using More Than One Rule at Once
Simplify each of the following, removing any negative exponents as needed.
 L4T 8 
(1)
 1 16 4 
16 L T 
(2)
2
(3)
 (2 xz 3 ) 4 ( x  2 z ) 2

(2 xz 2 ) 3

1
R 3 r 4V 2 
2



5
Answers
16 L16
16 L164 16 L12

 12
L4T 4T 8
T 4 8
T
6 8
4R r
V4
5
 16 x 4 z 12 x  4 z 2 


8x 3 z 6


(1)
(2)
(3)
 2 x 0 z 10 
=  x 3 z 6 


50
32 z
= x15 z 30
32
= 15 80
x z
5
Some applications of the exponents are as follows





Compound interest
Half life of radioactive decay
Bacteria multiplying
Probability of an event.
Scientific notation

Computers - Bits are used to measure storage capacity in computers - 1 byte =
Rational Exponents
Definition:
 
𝑛
If m and n are positive integer, we define am/n = √𝑎𝑚 = n a
Assuming a ≥ 0, if n is even.
Note: there are three different ways to write rational exponent
1.
2.
If m and n are negative integers, we define a-m/n = 𝑛
1
√𝑎𝑚
.
m
bits
Examples
1.
43/2 = √43 = √64 = 8
2.
1252/3 = √1252 =
3
3.
49

1
2
 125 
2
3
 5 2 = 25
1
1
1 1
 1 2
  


49
49 7
 49 
Radicals
𝑛
A radical is an expression of the form √𝑎 which denotes the principal nth root of a.
Where ‘a’ is Radicand, ‘n’ is the Index, ' ' is the Radical sign.
Radicals are also called Surd.
Note: n a  n a
𝑛
√𝑎 is negative if a is negative and n is odd.
Basic Laws of Radicals
n
n
n
1. Product Law: √ab = √a ∙ √b
n
a
2. Quotient Law: √b =
n
√a
n
√b
b≠0
n
n
( √ a) = a
3.
n m
√ √a = nm√a
4.
Examples of Laws
Product Law
𝑛
𝑛
𝑛
√𝑎𝑏 = √𝑎 ∙ √𝑏
1. 48  16  3  16 3  4 3
2. 30a 4  30 a 4  a 2 30
Quotient Law
𝑛
𝑎
𝑛
√𝑏 =
1.
√𝑎
𝑛
√𝑏
b≠0
32
32
16  2
16 2 4 2




25
5
25
25
25
2.
48
 16  4
3
𝑛
𝑛
Law3: ( √𝑎) = a
 7
8
8
7
𝑛
𝑚
Law 4: √ √𝑎 =
3 4
𝑛𝑚
√𝑎
2  12 2
Addition and Subtraction of radicals
Rule: Combine the coefficient of like radicands.
6 7 5 7 3 7  8 7
Multiplication of radicals
Rule: Multiply the numbers and multiply the radicands using the law of exponents and then
simplify the remaining radicals
Example:
Perform the indicated operation and express your answer in the simplest radical form utilizing
the Properties of Exponents and Radicals.
3
18a 2 b.3 3a 2 b.3 ab 2
Solution:
3
54a 221b112 =
3
54a 5b 4  3ab3 a 2 b
Division of radicals
Rule: Divide the number and divide the radicands using the law of exponents if possible and
then simplify the remaining radicals.
Example
56

7
8
4*2  2 2
Rationalisation of the denominator of the radical
Rule: Rationalizing the denominator means to remove any radicals from the
denominator.
Example
3
3
9
9 22
3
3
√9/2 = √2 = √2 (22 ) = √
9(22 )
23
3
=
√36
2
Review questions
Simplify the following radicals:
1.
3
x6 y 4
20
5
250  50  15 2
2.
3.
2
7
2 50  3 32
5.
Answers
4.
1.
2.
3.
4.
5.
3
x  y y  x 
2 3
3
3
2 33
y 3 3 y  x 2 y3 y
20
 4 2
5
25.10  25.2  15 2
5 10  5 2  15 2  5 10  10 2
2
=
7
2 7
14
.

7
7 7
2 50  3 32  2 25  2  3 16  2
= 2 5 2  3 4 2
= 10 2  12 2 =  2 2
Note: If radicands are NOT same, split into perfect and imperfect radicand to combine as
shown above.
Points to remember:
Definition of
exponent
Fundamental
operations of
Radicals
Addition/
subtraction
Formula/Concept
Exponents are shorthand for repeated multiplication of the same thing
by itself.
If n is a positive integer, an represents the product of factors, each of
which is a
𝑛
A radical is an expression of the form √𝑎 which denotes the principal
nth root of a.
n an a
𝑛
√𝑎 is negative if a is negative and n is odd.
Add up or Subtract the coefficient of the radical provided same
radicand in the expression.
Multiplication
Multiply the numbers first and multiply the radicands using the law of
exponents secondly and then simplify the remaining radical.
Division
Divide the number first and divide the radicands using the law of
exponents secondly if possible and then simplify the remaining
radicals.
Rationalizing the
denominator
Rationalizing the denominator means to remove any radicals from the
denominator.