# Simplifying Algebraic Expressions Multiplying and Dividing Contributed by: This pdf includes the following topics:-
Simplifying Equations
What is algebra
Equation
Operator
Associative Property
Commutative property
Distributive property
Multiplication
Division
1. Maths Refresher
Simplifying Equations
2. Simplifying Equations
Learning intentions ….
• Algebra
• Order of operations
• Commutative property
• Associative Property
• Distributive property
• Simplify with grouping symbols
3. What is algebra
• Algebra has many definitions; some adults have grown
up believing that algebra is somewhat scary and difficult.
• Algebra involves finding and communicating number
patterns and relationships
• As number patterns become more complex they are
more difficult to communicate verbally…
• Hence, notation is used to simplify the task.
• Algebra is actually a very useful and simple concept, the
complex part is familiarising yourself with the language.
• The simplest definition for algebra is:
– A mathematical method for finding an unknown
number.
4. • Equation: Is a mathematical sentence. It contains an equal sign
meaning that both sides are equivalent.
• Expression: An algebraic expression involves numbers,
operation signs, brackets/parenthesis and pronumerals that
substitute numbers.
• Operator: The operation (+ , − ,× ,÷) which separates the
terms.
• Term: Parts of an expression separated by operators.
• Pronumeral: A symbol that stands for a particular value.
• Variable: A letter which represents an unknown number. Most
common is 𝑥𝑥, but it can be any symbol.
• Constant: Terms that contain only numbers that always have
the same value.
• Coefficient: Is a number that is partnered with a variable.
Between the coefficient and the variable is a multiplication.
Coefficients of 1 are not shown.
5.
6. Glossary example
7. Some algebra rules …
Expressions with zeros and ones
• Zeros and ones can be eliminated, why:
• When we add zero it does not change the number,
𝑥𝑥 + 0 = 𝑥𝑥
• If we multiply by one, then the number stays the
same, for example: 𝑥𝑥 × 1 = 𝑥𝑥
• What we do to one side we do to the other
• …and the BODMAS rule
8. Order of Operations
Revision: Revision: Revision:
Example 1 Example 2 Example 3
50 − 3 × 2 × 5 = 10 + 2 − 3 × 4 = 32 ÷ 2 − 2 × 3 =
50 − 6 × 5 = 10 + 2 − 12 = 16 − 2 × 3 =
50 − 30 = 12 − 12 = 16 − 6 =
20 0 10
10.
11. Some algebra rules
• Multiplicative Property: 𝟏𝟏 × 𝒙𝒙 = 𝒙𝒙
• Multiplying any number by one makes no difference.
• Additive Inverse: 𝒙𝒙 + −𝒙𝒙 = 𝟎𝟎
• Any number added to its negative equals zero.
𝟏𝟏
• Multiplicative Inverse: 𝒙𝒙 × = 𝟏𝟏
𝒙𝒙
• Any number multiplied by its reciprocal equals one.
• Symmetric Property: 𝒙𝒙 = 𝒚𝒚 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒚𝒚 = 𝒙𝒙
• Perfect harmony!
• Transitive Property: 𝑰𝑰𝑰𝑰 𝒙𝒙 = 𝒚𝒚 𝒂𝒂𝒂𝒂𝒂𝒂 𝒚𝒚 = 𝒛𝒛 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝒛𝒛
• For example, if apples cost \$2 and oranges cost \$2
then apples and oranges are the same price.
12. Understanding an algebraic
Let’s investigate this pattern
• The first thing we do is number each element, if
you do not number the element you will not be
able to see the relationship between the ‘element
number’ (which is the ‘term’) and the ‘total number
of sticks’ in each element. If you cannot see the
relationship, how the pattern changes, you will
not be able to work out a formula.
13. Consider: Number of Total number
elements or of sticks
a)What is changing in this term
T1 4
b) What is the repeating part?
c) What stays the same? T2 7
• Each time it grows by 3 sticks. T3 10
T4 13
• The first stick remains the Tn y
same.
14. • Each new term grows by three, so for each term
– step in the pattern – another ‘group’ of three is
• However, there is always one matchstick that
stays the same – ‘the constant’
• Therefore, the generalisation (general rule) or
‘algebraic equation’ for the matchstick pattern
would be:
– nx3+1=y
– 3n + 1 = y
• (The number of elements (the term) times three
plus one = the total number of matchsticks)
15. Commutative property
• Think of the term ‘commutative’ in relation to being able
to move things around – to commute.
• Hence, Commutative Property is the property where we
can move things around
– The Commutative Law of Addition:
𝒙𝒙 + 𝒚𝒚 = 𝒚𝒚 + 𝒙𝒙
For example, 2+3 = 3+2
– The Commutative Law of Multiplication:
𝒙𝒙 × 𝒚𝒚 = 𝒚𝒚 × 𝒙𝒙
For example, 2 × 3 = 3 × 2
16. Associative Property
• The Associative Law of Addition:
𝒙𝒙 + 𝒚𝒚 + 𝒛𝒛 = 𝒙𝒙 + (𝒚𝒚 + 𝒛𝒛)
The order you add numbers does not matter. The
difference is that we ‘regroup’ the numbers
• The Associative Law of Multiplication:
𝒙𝒙 × 𝒚𝒚 × 𝒛𝒛 = 𝒙𝒙 × (𝒚𝒚 × 𝒛𝒛)
The order you multiply numbers does not matter.
The difference is that we ‘regroup’ the numbers,
whereas in commutative property the numbers are
moved around – not regrouped.
=
17. Distributive property
• The Distributive Law: multiplication distributes over
addition or subtraction through the brackets
(parentheses) 𝒙𝒙 𝒚𝒚 + 𝒛𝒛 = 𝒙𝒙𝒙𝒙 + 𝒙𝒙𝒙𝒙
For example, 2 3 + 4 = 2 × 3 + 2 × 4
2 7 =6+8
14 = 14
19. • Rewrite 3 × 2 × 𝑥𝑥 by using the
‘commutative property’
𝟑𝟑 × 𝟐𝟐𝟐𝟐 𝒐𝒐𝒐𝒐 𝟐𝟐 × 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 (simplified).
• Rearrange 2(4𝑥𝑥) in using the
‘associative property’
8 × 𝒙𝒙 𝟖𝟖𝟖𝟖 (simplified)
• Rewrite 8(2 + 𝑥𝑥) using the ‘distributive
property’
𝟖𝟖 × 𝟐𝟐 + (𝟖𝟖𝟖𝟖) 𝟏𝟏𝟏𝟏 + 𝟖𝟖𝟖𝟖 (simplified)
20. Collecting ‘like’ terms
Watch this short Khan Academy video for further explanation:
“Combining like terms, but more complicated”
expressions/v/combining-like-terms-3
21. Like terms
Often real life algebra problems look like the following:
• 7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 + 2 = 14
• It is difficult to even try to start solving a problem so large. What we
need to do is simplify the problem into a smaller problem. We do this
by collecting like terms.
• A like term is a term which has the same variable to the same
power only the coefficient is different.
• Looking at the example: 7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 all coefficients have
the same variable.
• We can treat them as a simple equation: 7 + 2 + 3 − 6 which equals
6, so:
7𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6𝑥𝑥 = 6𝑥𝑥
∴ 6𝑥𝑥 + 2 = 14
22. Like terms
Watch this short Khan Academy video for further explanation:
“Combining like terms and the distributive property”
expressions/v/combining-like-terms-and-the-distributive-property
23. Like terms
Collect the like terms and simplify:
5𝑥𝑥 + 3𝑥𝑥𝑥𝑥 + 2𝑦𝑦 − 2𝑦𝑦𝑦𝑦 + 3𝑦𝑦 2
Step 1: Recognise the like terms:
(𝑥𝑥𝑥𝑥 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑦𝑦𝑦𝑦 − 𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙)
5𝑥𝑥 + 3𝒙𝒙𝒙𝒙 + 2𝑦𝑦 − 2𝒚𝒚𝒚𝒚 + 3𝑦𝑦 2
Step 2: Arrange the expression so that the like terms are
together (Remember to take the operator with the term)
5𝑥𝑥 + 2𝑦𝑦 + 3𝒙𝒙𝒙𝒙 − 2𝒚𝒚𝒚𝒚 + 3𝑦𝑦 2
Step 3: Simplify
5𝑥𝑥 + 2𝑦𝑦 + 𝒙𝒙𝒙𝒙 + 3𝑦𝑦 2
25. a. 3𝑚𝑚 + 2𝑛𝑛 + 3𝑛𝑛 − 𝑚𝑚 − 7 = 2𝑚𝑚 + 5𝑛𝑛 − 7
a. 4 𝑥𝑥 + 7 + 3 2𝑥𝑥 − 2 = 4𝑥𝑥 + 28 + 6𝑥𝑥 − 6
= 10𝑥𝑥 + 22
c. 3 𝑚𝑚 + 2𝑛𝑛 + 4 2𝑚𝑚 + 𝑛𝑛 = 3𝑚𝑚 + 6𝑛𝑛 + 8𝑚𝑚 + 4𝑛𝑛
= 11𝑚𝑚 + 10𝑛𝑛
𝑥𝑥 𝑥𝑥 4𝑥𝑥+3𝑥𝑥 7𝑥𝑥
d. 3
+ =
4 12
=
12
26. Simplify with grouping symbols
Watch this short Khan Academy video for further explanation:
“Factoring and the distributive property 2”
algebra/v/factoring-and-the-distributive-property-2
27. Simplify with nested grouping
When there are two
sets of brackets –
one is nested inside
the other –
operations in the
inner set must be
worked first.
Watch this short Khan Academy video for further explanation:
“Expression terms, factors and coefficients”
terms-factors-and-coefficients
28. Simplify with nested grouping
Example 1 Example 2 Example 3
20 − [3 × 14 − 12 ]= 4[(6 + 3) × 10] = 3 + 24 =
20 − [3 × 2] = 4[9 × 10] = 12 − (10 − 7)
4(90) = 3+24
20 − 6 = =
12−3
14 360 27
=
9
3
30.
31. Simplify with grouping symbols
• Now let’s work with an algebraic
expression with brackets and simplify by
removing the brackets and including
powers.
• Recap:
• So if we multiply two numbers together, the
order in which we multiply is irrelevant –
commutative property
• Simplify 4 3𝑥𝑥
• This could be written as 4 × 3 × 𝑥𝑥
• And then as 4 × 3 × 𝑥𝑥
• Therefore, we can simplify to 12𝑥𝑥
32. Simplify with grouping
symbols and exponents
Another example incorporating exponents:
• Simplify (3𝑥𝑥)(6𝑥𝑥)
• 3 × 𝑥𝑥 × 6 × 𝑥𝑥
• We can change the order (3 × 6) × (𝑥𝑥 × 𝑥𝑥)
• Therefore, we can simplify to 18𝑥𝑥 2
33. Simplify with grouping symbols
and exponents
• Remember we need to follow the order
of operations rule BODMAS
• …and now we also apply the ‘INDEX
LAWS’ from the previous session
34. Simplify with grouping symbols
and exponents
Is 2𝑥𝑥 2 the same as (2𝑥𝑥)2 ?
• 2𝑥𝑥 2 is 2 × 𝑥𝑥 × 𝑥𝑥
– 𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑤𝑤𝑤𝑤𝑤𝑤 5
– 2 × 5 × 5 = 50
• (2𝑥𝑥)2 is 2 × 𝑥𝑥 × 2 × 𝑥𝑥
– 𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑤𝑤𝑤𝑤𝑤𝑤 5
– 10 × 10 = 100
35. Simplify with grouping symbols
and exponents
This example explains how any order property
can be used:
6𝑥𝑥 × 2𝑦𝑦 × 3𝑥𝑥𝑥𝑥 = 6 × 𝑥𝑥 × 2 × 𝑦𝑦 × 3 × 𝑥𝑥 × 𝑦𝑦
= 6 × 2 × 3 × 𝑥𝑥 × 𝑥𝑥 × 𝑦𝑦 × 𝑦𝑦
= 36𝑥𝑥 2 𝑦𝑦 2
36. Simplify with grouping symbols
and exponents
Simplify 5𝑥𝑥 2 × 6𝑥𝑥 5
• So we can say that we have
– (5 × 6) × (𝑥𝑥 × 𝑥𝑥) × (𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥)
– Which is 30 × 𝑥𝑥 2+5
– Therefore, 30𝑥𝑥 7
(Remember the first index law from last week.)
37. Simplify with grouping symbols
and exponents
• Simplify
– What is the difference
– 6𝑥𝑥 5𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 6𝑥𝑥 + 5𝑥𝑥
30𝑥𝑥 2 11𝑥𝑥 (Here we add like terms)
• One more, are these the same:
– (−5𝑎𝑎2 ) −2𝑎𝑎
– 10𝑎𝑎3
• −5 × −2 = 10
• 𝑎𝑎2 × 𝑎𝑎1 = 𝑎𝑎3 (index law one)
• Therefore, (−5𝑎𝑎2 ) −2𝑎𝑎 is the same as 10𝑎𝑎3
39. 1. 3𝑥𝑥 + 6𝑥𝑥 + 11𝑥𝑥 = 20𝑥𝑥
2. 3𝑥𝑥𝑥𝑥 + 8𝑥𝑥𝑥𝑥 = 11𝑥𝑥𝑥𝑥
3. 5𝑥𝑥 2 − 4𝑥𝑥 2 = 𝑥𝑥 2
4. 5𝑥𝑥 2 + 6𝑥𝑥 +4x = 5𝑥𝑥 2 +10𝑥𝑥
5. 7𝑥𝑥 2 y + 3𝑥𝑥 2 𝑦𝑦 + 6𝑥𝑥𝑥𝑥 = 10𝑥𝑥 2 𝑦𝑦 + 6𝑥𝑥𝑥𝑥
40. Simplify systematically
1. Simplify expressions that have grouping
symbols first and work from the innermost to
the outer. As you do this apply the BODMAS
rule too.
2. Simplify powers
3. Multiply in order from left to right
4. Add and subtract in order from left to right.
5. Then work backwards to check
42.
Simplifying expression with grouping symbols
and exponents
…and a challenge:
What is the missing number?
4( +3)2
2 =16
5(14−3 )
4( +3)2
• =16
5(14−9)
4( +3)2
• =16
5(5)
4( +3)2
• =16 so what divided by 25 = 16, 25 × 16 = 400
25
• Now we look at what divided by 4 = 400 which is 100
• What is the square root of 100? 10
• So now we can say that the missing number is 7
• Work backwards to see if this is correct
4( 7 +3)2
• =16
5(14−9)
45. Simplifying Equations
Reflect on the learning intentions ….
• Algebra
• Order of operations
• Commutative property
• Associative Property
• Distributive property
• Simplify with grouping symbols
46. Australian Mathematical Sciences Institute. (2011). Algebraic