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

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Answer 1 · 2017-10-04T23:26:40

The simplified result is #-1/(8m^2)#

We need two relations involving exponents:

#a^x*a^y=a^(x+y)#

and

#a^x -: a^y = a^(x-y)#

The three expressions in the numerator are all multiplied together, so we not that:

the two negative signs cancel each other
we can change the order of the powers to make the multiplication easier. This is called the commutative property, and it basically says #2xx3 = 3xx2# in simplest form.

So, the numerator becomes

#(2*2*2*m*m^2*m*n^(-3)*n^2*n^4) = 8*m^4*n^3#

Next combining this with the denominator, we have:

#(8*m^4*n^3)/(m^2*n^3)=8*m^(4-2)*n^(3-3)=8m^2#

(noting that #n^0# = 1)

Now that the large fraction has been simplified, note that the negative sign makes the whole thing negative:

#-8m^2#

Finally, the exponent of -1 applied to the entire bracket means the final answer is the reciprocal of the above expression.

Final answer: #-1/(8m^2)#