Question

How do you simplify $ {\left[ {{{\left( { - 2x{y^2}} \right)}^3}} \right] ^2} $ ?

Answer 1

Hint: To simplify this question , we need to solve it step by step . Starting form the parentheses with exponent over it , this means that the outer parentheses will be solved first and then the inner parentheses . The number $ - 2x{y^2} $ the number which can be expressed as having a cube over it raised to another exponent 2 also . We should solve it by using the basic formula of an exponent being raised to another exponent , we can multiply and write them for an equal value . Then simplify it to get the desired answer .

Complete step-by-step answer:
The number with whole exponent as cube can be expressed as the number with exponent by writing their respective cube that is for any number also for the number having exponent on exponent will be solved as follows = $ {\left( {{p^n}} \right)^n} = {p^{n \times n}} $ , p is number base and n is the exponent which is multiplied to the exponent over it .
 So , calculating the cube of the respective numbers = $ {\left[ {{{\left( { - 2x{y^2}} \right)}^3}} \right] ^2} $
Applying the above principle $ {\left( {{p^n}} \right)^n} = {p^{n \times n}} $ , we can say that,
 \[{\left( { - 2x{y^2}} \right)^{3 \times 2}}\]
 $ {\left[ {{{\left( { - 2x{y^2}} \right)}^3}} \right] ^2} = {\left( { - 2x{y^2}} \right)^{3 \times 2}} = {\left( { - 2x{y^2}} \right)^6} $
So , calculating the cube of the respective numbers = $ {\left( { - 2x{y^2}} \right)^6} $
 \[
   = {( - 2)^6} = 64 \\
   = {({y^2})^6} = {y^{2 \times 6}} = {y^{12}} \\
   = {(x)^6} = {x^6} \;
 \]
And then combining all the respective cubed terms to make our question solved = $ 64{y^{12}}{x^6} $ .
This is also the simplified solution to the question .
Therefore , the reduced form is $ 64{y^{12}}{x^6} $ $ $ . It is already in its simplest form .
So, the correct answer is “ $ 64{y^{12}}{x^6} $ $ $ ”.

Note: You should always remember that the square of -1 or any negative number will become positive .
You should always remember that the even exponent raised to -1 will become positive .
For example = any multiple of the number 2 in the exponent always gives the positive number . $ {( - 1)^{2 \times n}} = + ve $
But the cube of the negative number will always remain negative .
Greatest common factor itself describes the number which has a common factor , there is only one common factor that means the number is already simplified.