Question

How do you simplify `[(5-3) * (4-6)]^3 \div 8 + 3^4 \div 3^4 \div 3^2 - (2*3)^2 \div 12`?

Answer 1

See a solution process below:

Explanation:

Using the PEMDAS method, first execute all of the operations within Parenthesis:

`[(color(red)(5) - color(red)(3)) * (color(blue)(4) - color(blue)(6))]^3 -: 8 + 3^4 -: 3^4 -: 3^2 - (color(orange)(2) * color(orange)(3))^2 -: 12`

`[color(red)(2) * color(blue)(-2)]^3 -: 8 + 3^4 -: 3^4 -: 3^2 - 6^2 -: 12`

Next, execute the operation within the brackets which is another form of Parenthesis:

`(-4)^3 -: 8 + 3^4 -: 3^4 -: 3^2 - 6^2 -: 12`

Then execute the Exponent operations:

`(-4)^color(red)(3) -: 8 + 3^color(red)(4) -: 3^color(red)(4) -: 3^color(red)(2) - 6^2 -: 12`

`-64 -: 8 + 81 -: 81 -: 9 - 36 -: 12`

Next, execute the Division operations from left to right:

`(color(red)(-64) -: color(red)(8)) + ((color(blue)(81) -: color(blue)(81)) -: color(blue)(9)) - (color(orange)(36) -: color(orange)(12))`

`-color(red)(8) + (color(blue)(1) -: color(blue)(9)) - 3`

`-color(red)(8) + 1/9 - 3`

Now, execute the Addition and Subtraction from left to right:

`(9/9 xx -8) + 1/9 - 3`

`-72/9 + 1/9 - 3`

`-71/9 - 3`

`( -71/9) - (9/9 xx 3)`

`-71/9 - 27/9`

`-98/9`

If we want to write this as a mixed numbers we can convert as follows:

`-10 8/9`

Answer 2

`-10 8/9`

Explanation:

In an expression which has different operations, count the number of TERMS first. Terms are separated by `+ and -` signs

There are 3 terms in:

`color(red)([(5-3) * (4-6)]^3 \div 8)color(blue)( + 3^4 \div 3^4 \div 3^2)color(green)( - (2*3)^2 \div 12)`

Within each term, the order is

• brackets
• powers and roots
• multiply and divide

`" "color(red)([(5-3) * (4-6)]^3 \div 8)color(blue)( + (3^4 \div 3^4) \div 3^2)color(green)( - (2*3)^2 \div 12)`
`color(white)(xxxx)darrcolor(white)(xxxxx)darrcolor(white)(xxxxxxxx)darrcolor(white)(xxxxxxxx)darr`
`=" "color(red)([(2) *" " (-2)]^3 \div 8)color(blue)( " "+ 1" " \div 3^2)color(green)( " "- (6)^2 \div 12)`
`color(white)(xxxxxxxx)darrcolor(white)(xxxxxxxxxxxxxxx)darrcolor(white)(xxxx)darr`
`=color(white)(xxxx)color(red)([(-4)]^3color(white)(xxx) \div 8)color(blue)( " "+ 1" " \div 9)color(green)( " "- 36 \div 12)`
`color(white)(xxxxxxxx)darrcolor(white)(xxxxxxxxxxxxxx)darrcolor(white)(xxxxxxxx)darr`
#=color(white)(xxxxx)color(red)(-64color(white)(xxxxx) \div 8)" "color(blue)( " "+ 1/9)color(white)(xxxxxx)color(green)(- 3)#
`color(white)(xxxxxxxxxxx)darrcolor(white)`
#=color(white)(xxxxxxxx)color(red)(-8)color(white)(xxxxx)" "color(blue)( " "+ 1/9)color(white)(xxxxxx)color(green)(- 3)#

Each term has been simplified to a single value.

Re-arrange with the positive terms at the beginning:

`=1/9-8-3`

`-10 8/9`