Radical and exponents (2)

Chapter 3
Radical and
exponents

Exponential notation
 represent as to the th power .

Exponent
Base (integers)
(real number)

General case Special cases
(n is any positive integers)

Zero and negative exponent Example
(where a c ≠ 0)

Law of exponents
Law Example

Theorem on negative exponents

Prove:

Prove:

Example :
simplifying negative exponents

(1) 1 4 3 2
( x y )
3
1 2 4 2 3 2
( ) (x ) ( y )
3
2 8 6
3 x y
6
y
9 8
x

Principal nth root
Where n=positive integer greater than 1
= real number
Value for Value for

= positive real number b

Such that
=negative real number b

Such that

Properties of
RADICAL
index
radicand
Radical sign

PROPERTY EXAMPLE

Example:
combining radicals
Question: 4
α
3
α2
1
α 4
2
3
α
( 1 2 ) 5
α 4 3
α 12

1
5
α 12

1
12
α5

Law of radicals
law example

WARNING!

Example:
Removing factors from radicals

Question:

3a 2 b 3 6 a 5b
3a 2 b 3 .2.3a 5b
(32 a 6 b 4 )(2a )
3 2 2
(3a b ) ( 2a )
(3a 3b 2 ) 2 2a
3a 3b 2 2a

Rationalizing a denominator

Factor in denominator Multiply numerator Resulting factor
and denominator by

How do you know when a
radical problem is done?
(1) No radicals can be simplified.
Example:
8
(2) There are no fractions in the radical.
Example: 1
4
(3) There are no radicals in the denominator.
Example: 1
5

Example :
Rationalizing denominators

(1)

=

(2)

=

Definition of rational
exponents
m/n = rational number
n = positive integer greater than 1
= real number, then

(1)

(2)

(3)

Example:
Simplifying rational powers

(1)

6 4
x y
6 4
2 2
x .y
3 2
x y

How do you simplify variables in the radical?
7
x
Look at these examples and try to find the pattern…
1
x x What is the answer to x7 ?
2
x x x 7
x 3
x
3
x x x
4 2
As a general rule, divide the
x x exponent by two. The
x 5 2
x x remainder stays in the
6 3 radical.
x x

Definition of
• The logarithms of with base is defined by:
exponent

if and only if

base

For every and every real number .

Illustration
Logarithmic form Exponential form

• The logarithmic function with base is one-to-
one. Thus, the following equivalent conditions
are satisfied for positive real number x1 and x2 .

(1) If x1 x2 , then .
(2) If , then x1 x2 .

Example :
Solving a logarithms equation.

Check..

Since is a true statement, then

• Definition of common logarithm:

for every

• Defition of natural logarithm:

for every

Properties of logarithms
Logarithms with base Common logarithms Natural logarithms

= Power to which you need to raise 2 in order to get 8
(a) log28
= 3 ( Since 23 = 8 )

= Power to which you need to raise 4 in order to get 1
(b) log41
= 0 ( Since 40 = 1 )

= Power to which you need to raise 10 in order to get 10,000
(c) log1010,000
= 4 ( Since 104 = 10,000 )

= Power to which you need to raise 10 in order to get 1/100
(d) log101/100
= 2 ( Since 10-2 = 1/100 )

Laws of logarithms
Common logarithms Natural logarithms

Example:
Application law of logarithm

• log abc²
d3
= log (abc²) − log d 3
= log a + log b + log c² − log d 3
= log a + log b + 2 log c − 3 log d

Change of base formula
• If and if and are positive real
number, then

Special change of base formula

Example:
Solve

Solution :

Question 6

• Solve logb(x2) = logb(2x – 1).

x2 = 2x – 1
x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0

Then the solution is x = 1.

Question 7
• Solve ln( ex ) = ln( e3 ) + ln( e5 )
ln( ex ) = ln( e3 ) + ln( e5 )
ln( ex ) = ln(( e3 )( e5 ))
ln( ex ) = ln( e3 + 5 )
ln( ex ) = ln( e8 )
Comparing the arguments:
ex = e8
x=8

Question 8
Solve log2(x) + log2(x – 2) = 3

log2(x) + log2(x – 2) = 3
log2((x)(x – 2)) = 3
log2(x2 – 2x) = 3

23 = x2 – 2x
8 = x2 – 2x
0 = x2 – 2x – 8
0 = (x – 4)(x + 2)
x = 4, –2

Since logs cannot have zero or negative arguments, then the
solution to the original equation cannot be x = –2.
Solution: x=4

Question 10

If log10 5 + log10 (5x + 1) = log10 (x + 5) + 1,

SOLUTION :

log10 5 + log10 (5x + 1) = log10 (x + 5) + 1
log10 5 + log10 (5x + 1) = log10 (x + 5) + log10 10
log10 [5 (5x + 1)] = log10 [10(x + 5)]
5(5x + 1) = 10(x + 5)
5x + 1 = 2x + 10
3x = 9
x=3

Thank you…
Prepared by:
Nurul Atiyah binti Ripin
(D20111048011)
Irma Naziela binti Rosli
(D20111048007)

Radical and exponents (2)

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