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Ganit Learning Guides
Basic Algebra
Rational Exponents, Radicals and Equations
Author: Raghu M.D.
Contents
ALGEBRA .................................................................................................................... 2
RATIONAL EXPONENTS AND RADICALS ............................................................................................. 2
ALGEBRAIC EQUATIONS ..................................................................................................................... 6
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ALGEBRA
RATIONAL EXPONENTS AND RADICALS
In a term the number of times a variable x is multiplied by itself is called an exponent or
index.
x × x = x2 or square of x
x × x × x = x3 or cube of x
x × x × x × x = x4 or 4 th power (exponent) of x
x × x × x…………..m times = xm or mth power of x
x5 = x × x × x × x × x
x3 = x × x × x
Consider
and
x5 x × x × x × x × x
∴ 3 =
= x × x = x2
x× x× x
x
likewise
x m x × x × x.........mtimes
=
= x m −n
n
x × x...............ntimes
x
If m=n we have
xm xm
= n = x m− n = x 0 = 1
n
x
x
Also,
1 1 1
× × ............mtimes
x x x
=
1
x0
=
= x 0−1 = x − m
m
m
x
x
Laws of Exponents:
If x ≠ 0 and m and n are exponents
xm × xn = xm+n
= (xm)n = xm×n
(xy)m = xm × ym
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Exponents as Positive Rational Numbers:
⁄
⁄
, x3 and
are examples of exponents as rational numbers.
Any number that can be expressed as a quotient
is a rational number.
It is important to note that x can be a positive integer or quotient and the exponent can be any
integer or quotient.
If x is negative, it can still be simplified if the denominator of the exponent is an odd number.
For example: −8 ⁄ = −2
⁄
An exponent
can also be written as √
. It is called the radical form, where,
√ is the radical, x is the radicand, and √ is the radical sign.
Example 1: Find x, if x3 = 216
x = (216)
⁄
= √216 = 6
Answer: x = 6
Example2: Find x, if x = (81)3/2
x = (81) 3 / 2 =
( 81 ) = 9
3
3
= 729
Answer: x = 729
Example3: Show that
⁄
LHS =
×
⁄
×
⁄
= ⁄
=
= (
) =
= RHS
Example 4: Find x, if x = 0.64-3/2
x = 0.64
−3 / 2
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 64 
=

100 
−3 / 2
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=
1
 64 
100 


= −
3/ 2
( 100 )
( 64 )
3
100 3 / 2
=
=
64 3 / 2
=
1
64 3 / 2
100 3 / 2
3
10 3
1000
=
3
512
8
Answer: x =
1000
512
EXERCISE 2.1
1.
State the laws of exponents
2.
Find the value of x, if x equals:
a) 7
3.
4
b) 6
"
!
1/ 4
⁄
1/ 2
b) (3125)
⁄$
c) 27
1/ 3
 64 
×

 729 
2/3
Simplify:
 9  5 / 2 
a)   
 49  
7.
b) 216
 16 
c)  
 49 
−1 / 3
Simplify:
a) (343)
6.
!
c) √6561
b) √64
Find the value of x, if x equals:
a) (1296)
5.
3
Find the value of x, if x equals:
a) √343
4.
5
c)  
7
-3
Simplify:
a) (0.01)3/2
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3/5
b) (343)
b) (0.008)-4/3
−2 / 3
c) 27
1/ 3
 64 
×

 729 
2/3
c) (0.125)5/3
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8.
Simplify:
2
 27 64 
b) 
×

 125 343 
2 1/2
a) (6 + 8 )
9.
c) (15 x 5)2/3
⁄
Simplify and express in the form of
a) (
10.
2/3
)
c) (3 ) ÷ (2 ) b) √
Simplify:
a)(
b)
)
(
⁄
(
c)
*⁄+
⁄(
×
)
)
⁄
⁄+
×
⁄
Answers to EXERCISE 2.1
1.
As stated in the lesson
2.
a) 2401
3.
a) 7
b) 2
4.
a) 6
b)
5.
a) 2401
6.
a)
7.
a) 0.001
8.
a) 10
9.
a)
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27
b) c) 9
c)
,
*
b) 25
c)
,
b) 1 49
343
b) 625
b)
c) 16 27
c) 0.03125
b) 144 1225
⁄
$
c)
,
⁄
c) 25
c) 81 8
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10.
a) 1
b)
⁄
c)
⁄
ALGEBRAIC EQUATIONS
An equation indicates that two algebraic expressions are equal. A collection of terms is called
an expression. An algebraic equation relates to variables and numbers. If an equation is
having one variable, say x, then we need one equation to find the value of x.
However equations with higher powers of x, for example, x2 then x may have two or more
values.
+7,
Examples of expressions are:
−2
and so on.
Here x, 7 are called terms.
Example of an equation is:
+ 7 = 3 − 10
Properties:
Consider an equation ax + b = c, where a, b and c are constants. x is a variable, but it has
only one value which satisfies the equation. This number is known as a root or a solution to
the equation.
In an equation the value of x will not change if we carry out the following operations.
(1) Add or subtract the same number to both sides of the equation
For example 3 + 5 = 8:
3 +5+3= 8+3
3 + 8 = 11
Here 3x+8 is called the left-hand-side or LHS of the equation and 11 is called the
right-hand-side or RHS of the equation.
(2) Multiply or divide both sides of the equation by the same number
For example: 2 + 4 = 8:
1
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=
(
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+2=4
(3) A term can be transposed from LHS to RHS of an equation. However its sign will
change from positive to negative or negative to positive.
For example: 7 + 5 = 12
7 = 12 − 5 = 7
7 =7
Example 1:
Solve for x
3 + 7 = −2
3 = −7 − 2 = −9
+
= − = −3
= −3
Answer:
Example 2:
1 ,
1 +
= 3
=
By cross-multiplication:
(3 + 6) × 1 = (2 + 9) × 3
3 + 6 = 6 + 27
3 − 6 = 27 − 6
−3 = 21
=
Answer:
Basic-Algebra
= −7
= −7
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Example 3:
2x + 3 5
=
4x + 1 3
(2 + 3) × 3 = 5 × (4 + 1)
6 + 9 = 20 + 5
6 − 20 = 5 − 9
−14 = −4
=
= *
= Answer:
Example 4:
*
Sum of three successive integers is 21. Find the numbers.
Let the smallest number is x .
+ ( + 1) + ( + 2) = 21
Then
+
+1+
+ 2 = 21
3 + 3 = 21
3 = 21 − 3 = 18
=
(
= 6
The numbers are: 6, 6+1 and 6+2
Answer: 6, 7 and 8.
EXERCISE 2.2
1.
Solve 7x – 17 = 4
2.
Solve for x if
3.
Find x if
4.
Find x if
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( 1
)
( 1 *)
1
= 4
− = (
( 1
)
=4
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($1
)
= 2
5.
Solve for x:
6.
Find p if
52 + 17 = 3 − 22
7.
Find p if
52 = 152
8.
Find x if
9.
The temperature in degrees Kelvin is equal to the temperature in degrees Celsius plus
( 1
( 1
*
)
)
= −2
273°. Write an equation to convert the temperature from Celsius to Kelvin. Find the
temperature in degrees Kelvin if it is 27° degrees Celsius.
10.
Sum of three successive numbers is 33. Find the three numbers.
Answers to EXERCISE 2.2
1.
x=3
2.
= −5
3
=1
4.
=− 5.
=8
6.
2 = −2
7.
2=3
8.
= −6
9.
K = 273 + C, 300º K
10.
10, 11, 12
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