Question

Which is the simplified form of the expression

Answer 1

`\displaystyle A=( 4 )/( x^(14) y^(8))`

Explanation:

Properties of Powers

The algebraic expression called power has the form x^y where x is called the base and y is called the exponent.

Powers have some properties, some of which we'll recall below

`(x^a)^b=x^(ab)`

`(x^a)(x^b)=x^(a+b)`

`\displaystyle (x^a)/(x^b)=x^(a-b)`

`\displaystyle x^(-a)=(1)/(x^a)`

`\displaystyle x^(a)=(1)/(x^(-a))`

We'll use those properties to simplify the expression

`\displaystyle A=\left( ( (2)^(-3)(x^(-3)) (y^2) )/((4^(-2)) (x^4) (y^6)) \right)^2`

Taking the power 2 of every term

`\displaystyle A=( (2)^(-6)(x^(-6)) (y^4) )/((4^(-4)) (x^8) (y^(12)))`

Moving the terms with negative exponent to its opposite side

`\displaystyle A=( (2)^(-6) (4^(4))(y^4) )/( (x^8)(x^(6)) (y^(12)))`

Operating the explicit same bases

`\displaystyle A=( (2)^(-6) (4^(4)) )/( (x^(14)) (y^(12-4)))`

Since
`4^4=(2^2)^4=2^8`

`\displaystyle A=( (2)^(-6) (2^8) )/( (x^(14)) (y^(12-4)))`

Operating the remaining like bases subtractions

`\displaystyle A=( (2^2) )/( (x^(14)) (y^(8)))`

Finally

`\displaystyle A=( 4 )/( x^(14) y^(8))`