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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

what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn

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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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