Question

How do you simplify `(- \frac { 2n ^ { 4} \cdot - 2m n ^ { 4} \cdot 2m ^ { 0} n ^ { - 4} } { ( 2n ) ^ { 2} } ) ^ { 2}`?

Answer 1

`=4n^4m^2`

Explanation:

Simplifying the given power is determined by multiplying the
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powers, dividing them ,and finally squaring it by applying power of a power.
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Multiplication of powers is evaluated by multiplying the real
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numbers then adding exponents of powers having same base.
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`color(blue)(Ex: )`
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`color(blue)((-2a^3bxx3a)=(-2xx3)(a^3xxa^1)(b)=-6a^(3+1)b=-6a^4b)`
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Division of powers is evaluated by simplifying the real numbers
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then subtracting exponents of powers having same base.
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`color(red)(Ex: )`
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`color(red)((8n^4)/(16n^2)=(8/16)(n^4/n^2)=(1/2)(n^(4-2))=1/2n^2)`
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Power of a power is evaluated by multiplying the exponents.
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`color(green)(Ex: )`
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`color(green)((a^2)^3=a^(2xx3)=a^6)`

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`(-(2n^4.-2mn^4 .2m^0n^(-4))/(2n)^2)^2`
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`=(-(2n^4.-2mn^4 .2xx1xxn^(-4))/(2n)^2)^2" "`Substituting `" "m^0=1`
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`=((-(2xx-2xx2xx1)(n^4xxn^4xxn^(-4))m)/(4n^2))^2`
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`=(color(blue)(-(-8)(n^(4+4-4))m)/(4n^2))^2`
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`=((+8n^4m)/(4n^2))^2`
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`=(color(red)((8/4)(n^(4-2))m))^2`
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`=(2n^2m)^2`
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`=color(green)(2^2(n^2)^2m^2)`
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`=4n^4m^2`