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.
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.
i) Operation – Addition
Sign – “+”
a + b
In algebra we call it “sum” of two numbers.
ii) Operation – Subtraction
Sign – “-“
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 – “/”
In algebra we call it as a “quotient”.
When numbers are added or subtracted they are referred as Terms and when numbers are multiplied they are called Factors.s
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.
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
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
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
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
Let’s solve an expression to understand the rule better.
Solve 4+6(1+2)2-3 x 6/2
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
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
Although there is no such hard and fast rule between addition and subtraction, their order of operation can be reversed.
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
ii) x – (y + z)
= 5 – (10 + 2)
= 5 – 12
= 3 x 5
iv) x2 + y3 + z2
= 52 + 103 + 22
= 25 + 1000 + 4
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
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