Question

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

Answer 1

`64m^2n^4`

Explanation:

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

Apply the laws of indices.

`64m^2n^4`

Explanation:

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`