Intermediate Algebra (12th Edition)

$d=\dfrac{k^2}{F^2}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $F=\dfrac{k}{\sqrt{d}} ,$ in terms of $h ,$ square both sides. Then use the laws of exponents and the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Squaring both sides, the equation above is equivalent to \begin{array}{l}\require{cancel} (F)^2=\left(\dfrac{k}{\sqrt{d}}\right)^2 \\\\ F^2=\left(\dfrac{k}{\sqrt{d}}\right)^2 .\end{array} Using the extended Power Rule of the laws of exponents which states that $\left( \dfrac{x^m}{z^p} \right)^q=\dfrac{x^{mq}}{z^{pq}},$ the equation above is equivalent to \begin{array}{l}\require{cancel} F^2=\dfrac{(k)^2}{(\sqrt{d})^2} \\\\ F^2=\dfrac{k^2}{d} .\end{array} Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} d(F^2)=\left(\dfrac{k^2}{d}\right)d \\\\ dF^2=k^2 \\\\ d=\dfrac{k^2}{F^2} .\end{array}