CHAPTER8. EVALUATION OF EXPRESSIONS
8.1 Introduction: Expression is a term given to any mathematical
symbol. In algebra we use both, numbers and letters. Algebraic rules are
similar to the arithmetic rules; the difference is we use letters to represent
a general rule or general formula using letters, assuming it to be true for any
number.
For example: in arithmetic
7 + 3 = 3 + 7
In algebra a + b = b + a
Here a represents the first number and b represents the second number. a and b can be any number, satisfying the mentioned law.
8.2 Recognizing
an expression: An algebraic
expression is made up of signs and symbols of algebra. These symbols include
Arabic numeral, literal numbers etc. The numbers are
called the numerical symbols, and the letters are called literal symbols.
8.2.1 The
four Operations and their signs:
i) Operation
Addition
Sign +
a
+ b
In algebra we call it sum of two
numbers.
ii) Operation
Subtraction
Sign -
a
b
In algebra we call it as difference
between numbers.
iii) Operation
Multiplication
Sign x or . or *
a x b or a.b or a* b or simply ab
In algebra we call it as product of
numbers.
iv) Operation
Division
Sign /
a/b
In algebra we call it as a quotient.
8.4 Terms
and Factor: When numbers
are added or subtracted they are referred as Terms and when numbers are multiplied they are called Factors.
In algebra we
speak sum for of several terms even if there are subtractions in
it.
That is, find
the sum of the four terms
a -b + c d
Although the
expression above has subtraction in it yet we call it as sum of the given term.
8.5 Powers
and Exponents: When a
number is multiplied by itself repeatedly then we get the Power of that number.
a.a is called the second power
of a, a.a.a is called the third
power of a and so on.
In general a.a.a
..n times is called the nth power of a
Now,
a.a
= a2
a.a.a
= a3
a.a.a.a
= a4
a.a. to n times = an (where n can be any number)
a2 is
read as a squared and a3 is read as a cubed. The small numbers placed on a are called Exponents. In a2, 2 is the exponent, similarly in a3,
3 is the exponent
8.6 Function
of Parentheses:
Parentheses signify that what they enclose should be treated as one number.
4 + (4 x 3)
= 4 +12
5(7 + 2) = 5
x 9
9 + (1+ 6) =
9 + 7
10(3 x 2) =
10 x 6
Remember that if there is no sign between
a number and parentheses it means multiplication
8.7 Order
of operation: Order of
operation generally includes the BODMAS rule
That is Bracket or parentheses Open then Division then Multiplication
and then Subtraction
An algebraic
expression should be solved by following the above mentioned sequence.
Order of
operation
Step1. Evaluate
parentheses, in the expression first if
there are any.
Step2. Evaluate powers
or exponents.
Step3. Perform
Division
Step4. Perform
Multiplication
Step5. Perform Addition
Step6. Perform Subtraction
Lets solve an
expression to understand the rule better.
Solve 4+6(1+2)2-3 x
6/2
Solution
Step1.Evaluating
parentheses
4
+ 6 (3)2 3/6
Step2. Evaluating
exponents or power
4 + 6 x 9 3/6
Step3. Performing
division
4
+ 6 x 9 2
Step4. Evaluate the
product
4
+ 54 2
Step5. Evaluate the sum
58
2
Step6. Evaluate the
difference
56
The solution is 56
Taking another example with no parentheses and exponents
Solve 12 3 x 6 +
14/7
Start with division (Rule3 since there is no use of rule 1 and 2)
12 3 x 6 +2
Now the next step is multiplication
12 18 + 2
Now add
12- 20
Now subtract
-8 ans.
Although there is no such hard and fast rule between addition and
subtraction, their order of operation can be reversed.
8.7 Evaluation
of expression: Evaluation
of expression means replacing the letter with its value followed by the order
of operation.
Example: if x = 5, y = 10 and z = 2
Solve i) x + (y x z)
= 5 + (10 x 2)
= 5 + 20
= 25
ii) x (y + z)
= 5 (10 + 2)
= 5 12
= -7
iii) y(x/z)
= 3(10/2)
= 3 x 5
= 15
iv) x2
+ y3 + z2
= 52
+ 103 + 22
= 25 + 1000
+ 4
= 1029
8.8 Variable: A variable is a literal symbol and is so
called because its value may vary.
A variable, like x, is a blank or
empty symbol. And we can assign any value,
whether a positive number or a negative number, a whole number or a fraction.
Algebraic
expression with two variables: In these expressions value of one variable depends on the
value of other variable
y
= bx+c
Where b and c are constants, while x
and y are variables.
Example y
= 4x + 6
Here value of y depends on x
When x = 0, y = 4· 0 + 6 = 0 +6 =
6
When x = 1, y = 4· 1 + 6 = 4 + 6=
10
When x
= 2, y = 4· 2 + 6 = 8 + 6 =
14
When x = 3, y = 4· 3 + 6 = 12 + 6
= 18
When x = 4, y
= 4· 4
+ 6 = 16 + 6 = 22