To make [tex]\( A \)[/tex] the subject of the given relation:
[tex]\[
r = -100 \left( \left( \frac{A}{p} \right)^{1/n} - 1 \right)
\][/tex]
we will solve for [tex]\( A \)[/tex] step by step.
### Step 1: Simplify the Equation
First, we will divide both sides of the equation by [tex]\(-100\)[/tex]:
[tex]\[
\frac{r}{-100} = \left( \frac{A}{p} \right)^{1/n} - 1
\][/tex]
### Step 2: Isolate the Exponent Term
Next, we add 1 to both sides of the equation to isolate the term containing [tex]\( A \)[/tex]:
[tex]\[
\frac{r}{-100} + 1 = \left( \frac{A}{p} \right)^{1/n}
\][/tex]
### Step 3: Eliminate the Exponent
To remove the exponent [tex]\(\frac{1}{n}\)[/tex], we will raise both sides of the equation to the power of [tex]\( n \)[/tex]:
[tex]\[
\left( \frac{r}{-100} + 1 \right)^n = \frac{A}{p}
\][/tex]
### Step 4: Solve for [tex]\( A \)[/tex]
Finally, to solve for [tex]\( A \)[/tex], we multiply both sides of the equation by [tex]\( p \)[/tex]:
[tex]\[
A = p \left( \frac{r}{-100} + 1 \right)^n
\][/tex]
### Final Equation
The final equation, with [tex]\( A \)[/tex] as the subject, is:
[tex]\[
A = p \left( \frac{r}{-100} + 1 \right)^n
\][/tex]
By following these steps, we successfully isolated [tex]\( A \)[/tex] in the given relation.
