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Use of symbols
Objectives:
F Grade
Simplify expressions with one variable such as:
a + 2a + 3a
E Grade
Simplify expressions with more than one variable
such as:
2a + 5b + a – 2b
D Grade
Multiply out expressions with brackets such as:
3(x + 2) or 5(x - 2)
Factorise expressions such as 6a + 8 and x2 – 3x
C Grade
Expand (and simplify) harder expressions with
brackets such as:
x(x2 - 5) and
3(x + 2) - 5(2x – 1)
Algebraic Definitions
Expression
An expression is a mathematical "phrase" that stands
for a single number; for example, 3x + 1
Equation
An expression that equals something, that maybe another
expression or a single value.
For example:
3x + 1 = 7 or 3x + 1 = 2x - 1
Variable
A variable is a letter used in an algebraic expression in order
to represent any number.
Algebraic Definitions
Term
A term is a number and / or variable(s) connected with
x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation.
For example
3x + 4y
term
term
3x + 4y
a
b
term
term
Use of symbols
What combinations of letters and numbers mean
a+a
Can be read as 2 of those things called a
So this can be written as
2×a
because things get abbreviated in maths we write this as:
2a
a + a + a 3 of those things called a
3×a
So this becomes:
3a
Also, because multiplication is commutable (the order of the
Multipliers can be swapped and the answer remains the same)
a× b = b × a
or ab = ba
Use of symbols
This is not to be confused with:
a×a
Can be read as a of those things called a
So this can be written as
a×a
We know that 4 × 4 can be written as 42
because any number multiplied by itself like this
Index number
a × a = a2
So the following are also true:
a × a × a = a3
a × a × a × a = a4
32
Base number
The index number tells us how
many times the base number
is multiplied by itself.
e.g. 34 means 3 x 3 x 3 x 3 = 81
Use of symbols
Collecting terms of add & subtract
2a + 3a
a+a+a+a+a
So this can be simplified to 5 of those things called a
5a
7a - 3a
a+a+a+a+a+a+a-a-a-a
So this can be simplified to 4 of those things called a
5a
7b - 3a
b+b+b+b+b+b+v-a-a-a
So this cannot be simplified because a and b are different
This remains as:
7b - 3a
Use of symbols
Now do these:
1. p + 2p + 3p
6p
4. t + t + 4t
6t
7. d + 4d − 2d
3d
2. p + 4p − 3p
2p
5. f + 6f − 10f
– 3f
8. h + h − 5h + 2h
– 1h
3. 2ab + 3ab
5ab
6. 5ad − 2da
5ad
9. p2 + p2 + p2
3p2
Use of symbols
The index number of a letter or number only applies to the
number or letter immediately preceding it.
a3x = a × a × a × x
abc3 = a × b × c × c × c
ab3c = a × b × b × b × c
Mathematical convention is that where we have more than one
letter in a term, they are written in alphabetical order.
Use of symbols
Further rules for the use of letters
a × a = a2
a + a = 2a
These are different types of terms and cannot be mixed
If the index number in two terms is different they cannot be added
If the index number is the same they can be added / subtracted
Example:
Simplify this expression
5x2 + 2x + 2x2 –-5x
5x
The same power of x
these can be collected
The same power of x
these can be collected
7x2 - 3x
Use of symbols
Now do these:
1. x + 3x + 5 + x
5x + 5
4. y + 2 + y + 4
2y + 6
7. d + 5 + 2d − 3
3d + 2
2. g + 2g + h
3. 7ab + 2b + 4a + 2ab
3g + h
9ab + 4a + 2b
5. 7w + 6 − 2w + 2 6. p2+p2+2p+p2+4p
5w + 8
3p2+ 6p
8. 2x + 3y + 4x + 2y 9. 4w + 2 + y − 3
6x + 5y
4w + y - 1
10. a2 + a3 + 2a2
11. 4a2b + 5a2b 12. 3cd2+4cd2−2dc2−3c2d
9a2b
3a2+ 6a3
7cd 2- 5c 2 d
13. w5 + 2w5 + w
14. 5x3y + 2xy 3 + 2x 3 y
3w5+ w
7x3y + 2xy 3
15. ct 2 + 3t 2 + t − t 2
16. 7abc 2 + 4ab 2 c + 5abc 2 + 3ba 2 c
2ct2+ 2t2 + t
12abc 2 + 4ab 2 c + 3ba 2 c
Summary
F Grade
Use of symbols
Simplify expressions with one variable such as:
a + 2a + 3a = 6a
E Grade
Simplify expressions with more than one variable
such as:
2a + 5b + a – 2b = 3a + 3b
Algebraic meanings:
a + a + a = 3a
a× b = b × a or ab = ba
a × a × a = a3
•If the index number in two terms is different they cannot be added
•If the index number is the same they can be added / subtracted
•The index number of a letter or number only applies to the
number or letter immediately preceding it.
•Mathematical convention is that where we have more than one
letter in a term, they are written in alphabetical order.