How do you simplify #(-2a^7)(5a^-2)#?

1 Answer
May 4, 2016

Answer:

#-10a^5#

Explanation:

#color(red)("How you would normally see the calculation")#

variants on:

#color(red)(-(2xx5)(a^(7-2)) = -10a^5)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(green)("Very detailed solution split into 3 parts")#

Part 1 #-># signs
Part 2 #-># numbers
Part 3 #-># the letters (variables)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Part 1 "-> " signs")#

Write the signs as -1 and +1#" "# ( this does work!)

The signs are not the same as each other so

#(-1)xx(+1) = color(red)((-1))#

#color(blue)("So our answer is a negative value ")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Part 2 "-> " numbers")#

#2xx5 =color(red)(10)#
#color(blue)("The number part of the answer is 10")#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Part 3 "-> " letters (variables)")#

We have:#" "a^7xxa^(-2)#

#color(brown)("Shortcut method")#
Because the letters (variables) are the same we can do this:

#a^7xxa^(-2)" "=" "a^(7-2)" "=color(red)(a^5)#
'......................................................

#color(brown)("First principles method")#

#a^(-2)" is the same thing as "1/a^2#

So::#" "a^7xxa^(-2)" " =" " a^7xx1/a^2#

Let #a^7# be written as #a^5xxa^2#

Then we have: #a^5xxa^2xx1/a^2#

This is the same as
#" "a^5xxa^2/a^2#

But #a^2/a^2=1#

#" "a^5xxa^2/a^2" "=" "color(red)(a^5)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("putting it all together")#

#color(red)((-2a^7)(5a^(-2))= -10a^5)#