How do you multiply and simplify #\frac { ( - 4m ^ { 3} ) ^ { 2} ( n ^ { - 2} ) ^ { 2} } { ( 2^ { - 1} ) ^ { 2} m ^ { 4} n ^ { - 8} }#?

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Answer 1 · 2017-04-22T16:09:46

#64m^2n^4#

Consider just the numerator:

#(-4m^3)^2(n^(-2))^2" "->" "((-4)xx(-4)xxm^3xxm^3)/(n^2xxn^2) #
#" "=(16m^6)/n^4#

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Consider just the denominator:

#(2^(-1))^2m^4n^(-8)" "->" "1/2xx1/2 xxm^4xx1/n^8 #

#" "=m^4/(4n^8)#

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Putting it all together:

#(16m^6)/(n^4)-:m^4/(4n^8)#

#(16m^6)/(n^4)xx (4n^8)/m^4#

#16xx4 xx (m^4xxm^2)/m^4 xx (n^4xxn^4)/n^4#

#64m^2n^4#

Answer 2 · 2017-04-22T19:47:45

Apply the laws of indices.

#64m^2n^4#

There are several laws of Indices being applied in this fraction.

I will use the power law first to remove the brackets.
To do this, multiply the indices:

#(color(red)((-4m^3)^2)color(blue)((n^-2))^2)/(color(green)((2^-1)^2) color(purple)(m^4n^-8)#

#=(color(red)((-4)^2m^6)color(blue)((n^-4)))/(color(green)((2^-2) color(purple)(m^4n^-8)#

#=(color(red)(16m^6)color(blue)(xxn^-4))/(color(green)((2^-2) color(purple)(m^4n^-8)#

Recall: #x^-m = 1/x^m and 1/x^-n = x^n#

Get rid of the negative indices:

#=(color(red)(16m^6)xxcolor(green)(4)xxcolor(purple)(n^8) )/(color(purple)(m^4)color(blue)(xxn^4))#

Now simplify and subtract the indices of like bases.

#=64m^2n^4#