Question

Which is the simplified form of the expression

Answer 1

Answer:

$\displaystyle A=\frac{ 4 }{ x^{14} y^{8}}$

Step-by-step 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

$(x^a)^b=x^{ab}$

$(x^a)(x^b)=x^{a+b}$

$\displaystyle \frac{x^a}{x^b}=x^{a-b}$

$\displaystyle x^{-a}=\frac{1}{x^a}$

$\displaystyle x^{a}=\frac{1}{x^{-a}}$

We'll use those properties to simplify the expression

$\displaystyle A=\left( \frac{ (2)^{-3}(x^{-3}) (y^2) }{(4^{-2}) (x^4) (y^6)} \right)^2$

Taking the power 2 of every term

$\displaystyle A=\frac{ (2)^{-6}(x^{-6}) (y^4) }{(4^{-4}) (x^8) (y^{12})}$

Moving the terms with negative exponent to its opposite side

$\displaystyle A=\frac{ (2)^{-6} (4^{4})(y^4) }{ (x^8)(x^{6}) (y^{12})}$

Operating the explicit same bases

$\displaystyle A=\frac{ (2)^{-6} (4^{4}) }{ (x^{14}) (y^{12-4})}$

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

$\displaystyle A=\frac{ (2)^{-6} (2^8) }{ (x^{14}) (y^{12-4})}$

Operating the remaining like bases subtractions

$\displaystyle A=\frac{ (2^2) }{ (x^{14}) (y^{8})}$

Finally

$\displaystyle A=\frac{ 4 }{ x^{14} y^{8}}$