Transcript Numerical Expressions

Digital Lesson
Numerical Expressions
These are examples of numerical expressions.
2 (8  3)

18  [(10  2) 4]

2
32
56
54
4
3

9
grouping symbols
A numerical expression is an expression formed
from numbers by adding, subtracting, multiplying,
dividing, taking powers, taking roots, and using
grouping symbols: ( ), [ ], | |, { }, and the horizontal
bar as in fractions and radicals.
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To evaluate a numerical expression, carry out the indicated
operations to arrive at a single numerical value.
Examples: Evaluate.
1. 8 – 5(2) + 1 = 8 – 10 + 1
= –2 + 1
Value
= –1
2. – 2 + 3(7) = – 2 + 21
Value
= 19
35
3.
–1+5 =7–1+5
5
=6+5
= 11
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Value
3
To evaluate a variable expression at a given value, substitute the
variable with its value, then evaluate the numerical expression.
Examples: 1. Evaluate 2x + 3 when x = 5.
2(5) + 3 = 13
Value
2. Evaluate 3x2 + x – 4 when x = 2.
3(2)2 + (2) – 4 = 3 • 4 + (2) – 4
= 10
Value
( x  y) 2
3. Evaluate
when x = 3 and y = 1.
y
((3)  (1) ) 2 (4) 2 16
  16

(1)
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1
Value
1
4
Adding and Subtracting Signed Numbers
Examples: Add or subtract.
1. – 1 + (– 3) = – 4
-4 -3 -2 -1
0
The signs are the same, so add the
absolute values and attach the
common sign.
1
2
3
4
2. – 4 + 6 = 2
-4 -3 -2 -1
0
1
2
3
4
3. – 2 – (– 3) = – 2 + 3 = 1
-4 -3 -2 -1
0
1
2
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3
The signs are different, so subtract the
absolute values and attach the sign of
the number with the larger absolute
value.
To subtract, write the expression as the
addition of the opposite number.
4
5
Multiplying and Dividing Signed Numbers
Examples: Multiply or divide.
1. (–6) · (–2) = 12
The signs are the same, so the
product is positive.
2. (– 5) · 4 = – 20
The signs are different, so the
product is negative.
21
3.
 21
3
4.

Every division can be written as a
1




  7 multiplication problem.
 3
15
=3
5
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The signs are different, so the
quotient is negative.
The signs are the same, so the
quotient is positive.
6
The arithmetic of signed numbers and fractions can be used to
evaluate numerical expressions.
Examples: 1. Evaluate – x + 2x when x = – 5.
– (– 5) + 2(– 5) = –1 · (– 5) + 2 · (–5)
= 5 + 2 · (–5)
= 5 + (–10)
Signs are the same, so the
product will be positive.
Signs are different, so the
product will be negative.
Signs are different. Subtract.
= –5
 6x
2. Evaluate
when x = – 2.
x2
 6 (–2) 12

Division by zero is undefined.
(–2)  2
0
This expression is undefined when x = – 2.
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Adding and Subtracting Fractions
Examples: Add or subtract.
The denominators are the same,
so add the numerators.
Place the sum over the common
denominator.
The denominators are the same,
so subtract the numerators.
1 3 4
1.  
5 5 5
8 4 5
2.  
9 9 9
3 1 3 2 1
3.     
4 8 4 2 8
6 1
 
8 8
5

8

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Find a common denominator.
The common denominator is 8.
Subtract the numerators and place
the result over the denominator.
8
Multiplying and Dividing Fractions
Example: Multiply or divide.
4 1 4 ·1
1. · 
5 3 5 ·3
4

15
Multiply the numerators.
Multiply the denominators.
Place the product of the numerators over the
product of the denominators.
3  8  3  11 
2.        
5  11  5  8 

3  11

5 8
33

40


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The sign rules for fractions are the
same as those for integers.
Write as a multiplication by the
reciprocal of the divisor.
Multiply the numerators.
Multiply the denominators.
The result is negative since the signs
are different.
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The Order of Operations:
1. Perform all operations within grouping symbols as they
occur from the inside out. Grouping symbols can be
parentheses ( ), brackets [ ], the absolute value symbol | |,
and the horizontal bar used in fractions and radicals.
2. Simplify all exponential expressions.
3. Do all multiplications and divisions as they occur from
left to right.
4. Do all additions and subtractions as they occur from left
to right.
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Examples: Simplify.
1. 12 – 6 + 2 – 1 = 6 + 2 – 1
=8–1
Subtract.
Add.
=7
Subtract.
2. 3 · 5 – 2 = 15 – 2
Multiply.
= 13
3. 3 · (5 – 2) = 3 · 3
The grouping symbols
change the order of
operations, changing the
value of the expression.
=9
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Subtract.
Do subtraction within the
grouping symbols.
Multiply.
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Examples: Simplify.
1. 10 2  12  4  20  12  4
 20  3

 23
Multiply.
Divide.
Add.
2. 18  [(10  2)  4]2  18  [ 8  4]2 Evaluate expressions
 18  (2) 2
= 18 – 4
= 14
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inside grouping symbols
from the inside out.
Simplify the exponential
expression.
Subtract.
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Examples: Simplify.
1. 4 
54
60  6
2
3 4 
 32
3
2 1
54
4 
9
3
 4  18  9
Evaluate above and below the
fraction bar.
Simplify the exponential
expression.
Do left-most division.
 4 2
Divide.
2
Subtract.
2. 5  [(1  9 )]2 = 5 – [(1 + 3)]2 Square root
= 5 – (4)2
Do addition within grouping symbols.
= 5 – 16 = 11
Simplify exponent. Subtract.
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