Question

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

Answer 1

See a solution process below:

Explanation:

First, use this rule of exponents to eliminate the negative exponent in the numberator:

`x^color(red)(a) = 1/x^color(red)(-a)`

`2^color(red)(-1)/(2^3 * (2^2)^-4) => 1/(2^color(red)(- -1) * 2^3 * (2^2)^-4) => 1/(2^color(red)(1) * 2^3 * (2^2)^-4)`

We can next use this rule of exponents to get simplify the term with the outer exponent:

`(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))`

`1/(2^1 * 2^3 * (2^color(red)(2))^color(blue)(-4)) => 1/(2^1 * 2^3 * 2^(color(red)(2) xx color(blue)(-4))) => 1/(2^1 * 2^3 * 2^-8)`

Then, we can use this rule of exponents to combine the terms in the denominator:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`1/(2^color(red)(1) * 2^color(blue)(3) * 2^color(green)(-8)) => 1/(2^(color(red)(1) + color(blue)(3) + color(green)(-8))) = 1/2^-4`

Now, use this rule of exponents to complete the simplification:

`1/x^color(red)(a) = x^color(red)(-a)`

`1/2^color(red)(-4) => 2^color(red)(- -4) => 2^color(red)(4) => 16`