1 Answer

Jonny Buchanan · Answer 1 · 2024-11-24T14:35:12+0000

To simplify the expression
$\displaystyle\sf f(x/y) = \left((9)/(16)\right)^(-3) / \left(\left((9)/(16)\right)^(-2)\right)$ and find the value of
$\displaystyle\sf \left((x)/(y)\right)^(-1) + \left((y)/(x)\right)^(-1)$ , we can begin by evaluating each component separately.

Let's simplify
$\displaystyle\sf \left((9)/(16)\right)^(-3)$ :

$\displaystyle\sf \left((9)/(16)\right)^(-3)$ can be rewritten as
$\displaystyle\sf \left((16)/(9)\right)^(3)$ . Taking a number to the power of -n is equivalent to taking its reciprocal and raising it to the power of n:

$\displaystyle\sf \left((16)/(9)\right)^(3) = \left((9)/(16)\right)^(-3)$ .

So,
$\displaystyle\sf \left((9)/(16)\right)^(-3) = \left((16)/(9)\right)^(3)$ .

Next, let's simplify
$\displaystyle\sf \left((9)/(16)\right)^(-2)$ :

$\displaystyle\sf \left((9)/(16)\right)^(-2)$ can be rewritten as
$\displaystyle\sf \left((16)/(9)\right)^(2)$ :

$\displaystyle\sf \left((9)/(16)\right)^(-2) = \left((16)/(9)\right)^(2)$ .

Now, let's evaluate
$\displaystyle\sf \left((x)/(y)\right)^(-1) + \left((y)/(x)\right)^(-1)$ :

$\displaystyle\sf \left((x)/(y)\right)^(-1) + \left((y)/(x)\right)^(-1)$ is equivalent to
$\displaystyle\sf (1)/(\left((x)/(y)\right)) + (1)/(\left((y)/(x)\right))$ .

Simplifying this expression further:

$\displaystyle\sf (1)/(\left((x)/(y)\right)) + (1)/(\left((y)/(x)\right)) = (y)/(x) + (x)/(y)$ .

To simplify the overall expression
$\displaystyle\sf f(x/y) = \left((9)/(16)\right)^(-3) / \left(\left((9)/(16)\right)^(-2)\right)$ , we substitute the simplified values:

$\displaystyle\sf f(x/y) = \left((16)/(9)\right)^(3) / \left((16)/(9)\right)^(2)$ .