College Algebra (11th Edition)

$x=\left\{ 0,\dfrac{1}{81} \right\}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $3x^{3/4}=x^{1/2} ,$ raise both sides to the fourth power to get rid of the denominators in the fractional exponents. Then factor the resulting equation. Use the Zero Product Property by equating each factor to zero and then solving the equations separately. Finally, do checking if the solution satisfies the original equation. $\bf{\text{Solution Details:}}$ Raising both sides to the fourth power, the given equation becomes \begin{array}{l}\require{cancel} \left( 3x^{3/4} \right)^4=\left( x^{1/2} \right)^4 .\end{array} Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to \begin{array}{l}\require{cancel} 3^4x^{\frac{3}{4}\cdot4 }=x^{\frac{1}{2}\cdot4} \\\\ 81x^{3}=x^{2} \\\\ 81x^{3}-x^{2}=0 .\end{array} Factoring the $GCF=x^2,$ the equation above is equivalent to \begin{array}{l}\require{cancel} x^2(81x-1)=0 .\end{array} Equating each factor to zero (Zero Product Property), then \begin{array}{l}\require{cancel} x^2=0 \\\\\text{OR}\\\\ 81x-1=0 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} x^2=0 \\\\ x=0 \\\\\text{OR}\\\\ 81x-1=0 \\\\ 81x=1 \\\\ x=\dfrac{1}{81} .\end{array} Upon checking, $x=\left\{ 0,\dfrac{1}{81} \right\}$ satisfies the original equation.