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Algebra (Intermediate) - Simplifying Expressions
 
Expressions | Rules for Addition and Subtraction | Multiplication of Terms | Dividing Terms
 

Expressions

Expressions are groups of letters separated by + or – signs.

3p + 2t is an expression.

3p and 2t are called terms.

Like terms are the same letters.

 

Rules for Addition and Subtraction

Expressions can be simplified by adding or subtracting ‘like terms' only.

See how the following expressions may be simplified:

t + t + t = 3t
3t – t = 2t
4p + 3p = 7p
pq + pq = 2pq
q2 +q2 = 2q2

These all have ‘like terms' and can therefore be simplified (Note: the powers must also be the same).

The expressions below do not have like terms and so cannot be simplified:

3y + 2t = 3y + 2t
4y + 3 = 4y +3
y2 + y3 = y2 +y3
5x – 3y = 5x – 3y

This can be applied to more difficult expressions, as in the following examples.

Example 1: Simplify 3t + 4p + 2t - 3p

3t + 2t = 5t (Note: a term has a sign ‘in front of it' which must stay with it)

4p – 3p = p

Therefore 3t + 4p + 2t – 3p = 5t + p


Example 2: Simplify 5y + 6x – 3y – 8x

5y – 3y = 2y

6x – 8x = –2x

Therefore 5y + 6x – 3y – 8x = 2y – 2x

 
Multiplication of Terms
a. Like Terms
y × y x y = y3
  
 

y x y x y x y = y4

  

The small number is an index, commonly called a ‘power' — it tells us how many times to multiply a term by itself.

Example: p5 = p x p x p x p x p

p5 x p2 = p x p x p x p x p x p x p = p7

Note: this can be achieved by simply adding the powers as follows.

p5 x p2 = p5 + 2 = p7

See how these expressions have been simplified:


3p2 x 5p3 = 15p5

2y3 x 4y4 = 8y7

b. Unlike Terms

See how the following expressions have been simplified:

p x q = pq  (Note: we leave out the times sign).

3p x 2q = 6pq (Multiply the numbers first and then the letters.)

p2 x q3 = p2 q3

5 x q = 5q


Rules for Multiplying in Algebra

Rule 1: For ‘like terms' we add the powers.

Rule 2: For ‘unlike terms' we leave out the multiply sign.

 
Dividing Terms

a. Like Terms
Simplify the following:
t5 / t2 =
t5
(used as the divide sign in algebra)
  
   
t2
    
 
=
 t x t x t x t x t  
   
        t x t
    
 

 =

  t3    
Therefore
  t5 / t2=  t3  

This can be done by subtracting the powers, as in the example below.

6p 7 / 3p 2 = 2p 5
The numbers are divided first and then the letters.


b. Unlike Terms:
Example 1: simplify the following p5 / y3 = p5
    y3

In this case, we cannot subtract the powers.

Example 2: simplify the following 6q3 / 2t5 = 6q3
  2t5
   
 

  =

  3q3
   t5

In this case, we can divide the numbers.