Simplifying Algebraic Expressions
Letters (variables) are used to stand in the place of numbers that we don’t know. This
enables us (or others) to observe patterns and write rules (laws) and formulae to describe
situations. It also enables us to use other peoples rules or formulae to perform calculations
in a specific situation. Understanding basic algebra allows us to do these things.
Background Ideas
1.
In algebra the multiplication sign is left out.
2a = 2 x a
2.
abc = a x b x c
3(a + 4) = 3 x (a + 4)
The number in a group of letters (term) is the co-efficient and is always at the front.
3abc never a3bc
3.
a = 1a and -a = -1a (the co-efficient of 1 is not written)
First Question – What is the operation?
Operations are things we do to numbers – the four basic operations are addition, subtraction,
multiplication and division. The way we simplify algebraic expressions changes for different
operations.
Addition and Subtraction
1.
You can only add/subtract terms if the letters are exactly the same.
For example:
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2a + 3a
can add
(2a + 3a = 5a)
2a + 3b
cannot add
2ab + 3ac
cannot add
3x + 2x²
cannot add
5x² - 3x²
can subtract
(5x² - 3x² = 2x²)
5a²bc - 2a²bc
can subtract
(5a²bc - 2a²bc = 3a²bc)
2.
Add/subtract the co-efficients. The letter(s) does not change.
For example:
2a + 3a = 5a
5x² - 3x² = 2x²
5a²bc - 2a²bc = 3a²bc
3.
Every term grabs the sign on the left hand side of it.
3a
+12b
+5a
2x
-5y
+3x
3a
+5b
+a
-2b =
-8y
8a + 10b
+ 3w
=
5x - 13y + 3w
-b = 4a + 4b
If the answer is positive, add it
If the answer is negative, subtract it.
You need to be able to add/subtract negative numbers – if unsure check on calculator
Exercise 1:
Simplify the following
1. 2a + 5a
2. 6b - 4b
3. a + a
4. 6b - b
5. 3a + 2c
6. 3ab + 2ac
7. 3a + 4b + 2a - 3b
8. 3p + 2q - 2r - p + 5q
9. 3a 2b  2ab 2  a 2b  3ab 2
10. 4a - 3b - 6a - 2a + 2b + 10
Multiplication
1.
You can multiply any terms together.
2.
In multiplication the letter(s) do change.
3.
Multiply the co-efficients (numbers).
4.
Multiply letters by leaving out multiplication sign.
For example:
a x b = ab
3a x 2b = 6ab
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To simplify more complicated expressions involving multiplication we need to know some
index laws
Index Laws
1.
a x an  am  n
m
a2 x a4  a6
For example:
2.
a x a5  a6
and
(Note: a1  a )
(a m )n  a mn
For example: (a 2 )3  a 6
3.
(ab)
m
 a mb m
For example: (2ab) 4  2 4 a 4 b 4  16a 4 b 4
and
(3a 2b 3 )2  9a 4b 6
Putting ideas about multiplication and the index laws together
Examples:
3a 2b 5 x 2a 3b 3 c  6a 5b 8 c
a 2b x ab 3  a 3b 4
(2a 2b)3 x (3ab 4 )2  8a 6b 3 x 9a 2b 8  72a 8b11
Exercise 2:
1. 3a x 2b
2. 3bc x a
3. 2a 2b x 3ab 2
4. 3a 3b 5 x 4ab 6
5. 2a x 3a 2 x 4a 3
6. (a 2b 5 ) 4
7. (2a 2b)3
8. (3a 2b) 2 x (2ab) 4
Exercise 3:
1. 2a + 3a
2. 2a x 3a
3. a + a
4. a x a
5. 3a 2b  4a 2b
6. 3a 2b x 4a 2b
7. 3a 2b  5ab 2
8. 3a 2b x 5ab 2
9. (6ab 3 c 5 ) 2
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Answers
Exercise 1
5. can’t add
1. 7a
2. 2b
3. 2a
4. 5b
6. can’t add
7. 5a + b
8. 2p + 7q - 2r
9. 2a 2b  5ab 2
1. 6ab
2. 3abc
3. 6a 3 b 3
4. 12a 4 b11
5. 24a 6
6. a 8 b 20
7. 8a 6 b 3
8. 144a 8 b 6
1. 5a
2. 6a 2
3. 2a
4. a 2
5. 7a 2 b
6. 12a 4 b 2
7. can’t add
8. 15a 3 b 3
9. 36a 2b 6 c 10
10. -4a - b + 10
Exercise 2
Exercise 3
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