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Using the order of operations

Use the order of operations to evaluate each of the following expressions.

  1. ( 3 2 ) 2 4 ( 6 + 2 )
  2. 5 2 4 7 11 2
  3. 6 | 5 8 | + 3 ( 4 1 )
  4. 14 3 2 2 5 3 2
  5. 7 ( 5 3 ) 2 [ ( 6 3 ) 4 2 ] + 1

  1. ( 3 2 ) 2 4 ( 6 + 2 ) = ( 6 ) 2 4 ( 8 ) Simplify parentheses = 36 4 ( 8 ) Simplify exponent = 36 32 Simplify multiplication = 4 Simplify subtraction

  2. 5 2 4 7 11 2 = 5 2 4 7 9 Simplify grouping symbols (radical) = 5 2 4 7 3 Simplify radical = 25 4 7 3 Simplify exponent = 21 7 3 Simplify subtraction in numerator = 3 3 Simplify division = 0 Simplify subtraction

    Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.


  3. 6 | 5 8 | + 3 ( 4 1 ) = 6 | −3 | + 3 ( 3 ) Simplify inside grouping symbols = 6 3 + 3 ( 3 ) Simplify absolute value = 6 3 + 9 Simplify multiplication = 3 + 9 Simplify subtraction = 12 Simplify addition

  4. 14 3 2 2 5 3 2 = 14 3 2 2 5 9 Simplify exponent = 14 6 10 9 Simplify products = 8 1 Simplify differences = 8 Simplify quotient

    In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.


  5. 7 ( 5 3 ) 2 [ ( 6 3 ) 4 2 ] + 1 = 7 ( 15 ) 2 [ ( 3 ) 4 2 ] + 1 Simplify inside parentheses = 7 ( 15 ) 2 ( 3 16 ) + 1 Simplify exponent = 7 ( 15 ) 2 ( −13 ) + 1 Subtract = 105 + 26 + 1 Multiply = 132 Add
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Use the order of operations to evaluate each of the following expressions.

  1. 5 2 4 2 + 7 ( 5 4 ) 2
  2. 1 + 7 5 8 4 9 6
  3. | 1.8 4.3 | + 0.4 15 + 10
  4. 1 2 [ 5 3 2 7 2 ] + 1 3 9 2
  5. [ ( 3 8 ) 2 4 ] ( 3 8 )
  1. 10
  2. 2
  3. 4.5
  4. 25
  5. 26
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Using properties of real numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative properties

The commutative property of addition    states that numbers may be added in any order without affecting the sum.

a + b = b + a

We can better see this relationship when using real numbers.

( −2 ) + 7 = 5 and 7 + ( −2 ) = 5

Similarly, the commutative property of multiplication    states that numbers may be multiplied in any order without affecting the product.

a b = b a

Again, consider an example with real numbers.

( −11 ) ( −4 ) = 44 and ( −4 ) ( −11 ) = 44

It is important to note that neither subtraction nor division is commutative. For example, 17 5 is not the same as 5 17. Similarly, 20 ÷ 5 5 ÷ 20.

Associative properties

The associative property of multiplication    tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

a ( b c ) = ( a b ) c

Consider this example.

( 3 4 ) 5 = 60 and 3 ( 4 5 ) = 60

The associative property of addition    tells us that numbers may be grouped differently without affecting the sum.

a + ( b + c ) = ( a + b ) + c

This property can be especially helpful when dealing with negative integers. Consider this example.

[ 15 + ( −9 ) ] + 23 = 29 and 15 + [ ( −9 ) + 23 ] = 29

Are subtraction and division associative? Review these examples.

8 ( 3 15 ) = ? ( 8 3 ) 15 64 ÷ ( 8 ÷ 4 ) = ? ( 64 ÷ 8 ) ÷ 4 8 ( 12 ) = 5 15   64 ÷ 2 = ?   8 ÷ 4 20   20 10   32 2

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
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Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
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cos 18 ____ sin 72 evaluate
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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