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Key Words and Introduction to
Expressions
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Writing expressions
Here are some examples of algebraic expressions:
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n+7
a number n plus 7
5–n
5 minus a number n
2n
2 lots of the number n or 2 × n
6
n
6 divided by a number n
4n + 5
4 lots of a number n plus 5
n3
a number n multiplied by itself twice or
n×n×n
3 × (n + 4)
or 3(n + 4)
a number n plus 4 and then times 3.
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Writing expressions
Miss Green is holding n number
of cubes in her hand:
Write an expression for the number of cubes in her hand if:
She takes 3 cubes away.
n–3
She doubles the number of
cubes she is holding.
2n
or
2×n
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Equivalent expression match
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Simplifying Algebraic
Expressions
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Like terms
An algebraic expression is made up of terms and operators
such as +, –, ×, ÷ and ( ).
A term is made up of numbers and letter symbols but not
operators.
For example,
3a + 4b – a + 5 is an expression.
3a, 4b, a and 5 are terms in the expression.
3a and a are called like terms because they both contain a
number and the letter symbol a.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
5+5+5+5=4×5
In algebra,
a + a + a + a = 4a
The a’s are like terms.
We collect together like terms to simplify the expression.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
(7 × 4) + (3 × 4) = 10 × 4
In algebra,
7 × b + 3 × b = 10 × b
or
7b + 3b = 10b
7b, 3b and 10b are like terms.
They all contain a number and the letter b.
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Collecting together like terms
Remember, in algebra letters stand for numbers, so we
can use the same rules as we use for arithmetic.
In arithmetic,
2 + (6 × 2) – (3 × 2) = 4 × 2
In algebra,
x + 6x – 3x = 4x
x, 6x, 3x and 4x are like terms.
They all contain a number and the letter x.
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Collecting together like terms
When we add or subtract like terms in an expression we
say we are simplifying an expression by collecting
together like terms.
An expression can contain different like terms.
For example,
3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b
= 7a + 8b
This expression cannot be simplified any further.
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Collecting together like terms
Simplify these expressions by collecting together like terms.
1) a + a + a + a + a = 5a
2) 5b – 4b = b
3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6
= 2c + 2d + 9
4) 4n + n2 – 3n = 4n – 3n + n2 = n + n2
5) 4r + 6s – t
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Cannot be simplified
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Algebraic perimeters
Remember, to find the perimeter of a shape we add
together the lengths of each of its sides.
Write algebraic expressions for the perimeters of the following
shapes:
2a
Perimeter = 2a + 3b + 2a + 3b
3b
= 4a + 6b
5x
4y
x
5x
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Perimeter = 4y + 5x + x + 5x
= 4y + 11x
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Using Index Notation
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Using index notation
Simplify:
x + x + x + x + x = 5x
Simplify:
x to the power of 5
x × x × x × x × x = x5
This is called index notation.
Similarly,
x × x= x2
x × x × x = x3
x × x × x × x = x4
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Using index notation
We can use index notation to simplify expressions.
For example,
3p × 2p =3 × p × 2 × p = 6p2
q2 × q3 = q × q × q × q × q = q5
3r × r2 =3 × r × r × r = 3r3
2t × 2t =(2t)2
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or
4t2
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Expressions of the form (xm)n
When a term is raised to a power and the result raised to
another power, the powers are multiplied.
For example,
(a5)3 = a5 × a5 × a5
= a(5 + 5 + 5)
= a15 = a(3 × 5)
In general,
(xm)n = xmn
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Expressions of the form (xm)n
Rewrite the following without brackets.
1) (2a2)3 =
8a6
2) (m3n)4 = m12n4
3) (t–4)2 =
t–8
4) (3g5)3 = 27g15
5) (ab–2)–2 =
a–2b4
6) (p2q–5)–1 =
p–2q5
7) (h½)2 =
h
8) (7a4b–3)0 =
1
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The zero index
Look at the following division:
Any number or term divided
by itself is equal to 1.
y4 ÷ y4 = 1
But using the rule that xm ÷ xn = x(m – n)
y4 ÷ y4 = y(4 – 4) =y0
That means that
y0 = 1
In general, for all x  0,
x0 = 1
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Index laws
Here is a summary of the index laws.
xm × xn = x(m + n)
(xm)n = xmn
x0 = 1 (for x = 0)
x1 = x
x = x
1
2
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Multiplying Algebraic Terms
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Multiplying terms together
In algebra we usually leave out the multiplication sign ×.
Any numbers must be written at the front and all letters should
be written in alphabetical order.
For example,
4 × a =4a
1 × b =b
We don’t need to write a 1 in front of the letter.
b × 5 =5b
We don’t write b5.
3 × d × c = 3cd We write letters in alphabetical order.
6 × e × e = 6e2
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Brackets
Look at this algebraic expression:
4(a + b)
What do you think it means?
Remember, in algebra we do not write the multiplication sign, ×.
This expression actually means:
4 × (a + b)
or
(a + b) + (a + b) + (a + b) + (a + b)
=a+b+a+b+a+b+a+b
= 4a + 4b
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Expanding Algebraic
Expressions
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Expanding brackets then simplifying
Sometimes we need to multiply out brackets and then simplify.
For example,
2(5 – x)
We need to multiply the bracket by 2.
10 – 2x
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Expanding brackets then simplifying
Simplify
4 – (5n – 3)
We need to multiply the bracket by –1 and collect together
like terms.
4 – 5n + 3
= 4 + 3 – 5n
= 7 – 5n
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Expanding brackets then simplifying
Simplify
2(3n – 4) + 3(3n + 5)
We need to multiply out both brackets and collect together
like terms.
6n – 8 + 9n + 15
= 6n + 9n – 8 + 15
= 15n + 7
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Expanding brackets then simplifying
Simplify
5(3a + 2b) – 2(2a + 5b)
We need to multiply out both brackets and collect together
like terms.
15a + 10b – 4a –10b
= 15a – 4a + 10b – 10b
= 11a
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