Question

Find all real solutions of the equation.

$x^{\frac{1}{2}}-3x^{\frac{1}{3}}=3x^{\frac{1}{6}}-9$

Short answer

The real solutions of the given equation are$x=27$ and $x=729$.

Step-by-step solution

Step 1. Definition of equation.

Equation is a mathematical statement in which two expressions are equal to each other.

Step 2. Given Information.

The given equation is:

$x^{\frac{1}{2}}-3x^{\frac{1}{3}}=3x^{\frac{1}{6}}-9$

Step 3. Simplify the given equation.

Substitute$x=y^6$ in the given equation.

role="math" localid="1643913742602" $$\begin{gathered} {\left(y^6\right)}^{\frac{1}{2}}-3{\left(y^6\right)}^{\frac{1}{3}}=3{\left(y^6\right)}^{\frac{1}{6}}-9 \\ {y}^3-3y^2=3y-9 \\ {y}^3-3y^2-3y+9=0 \\ {y}^2\left(y-3\right)-3\left(y-3\right)=0 \\ \left(y^2-3\right)\left(y-3\right)=0 \end{gathered}$$

Step 4. Solve the above equation.

Use the zero product property.

$$\begin{gathered} y^2-3=0 \\ {y}^2=3 \\ y=\pm \sqrt{3} \\ {x}^{\frac{1}{6}}=\pm \sqrt{3}\left[\because y^6=x\Rightarrow y=x^{\frac{1}{6}}\right] \end{gathered}$$

The sixth root of a real number cannot be a negative number. So,$x^{\frac{1}{6}}=-\sqrt{3}$has no real solution.

localid="1643913952825" $\begin{array}{c}{x}^{\frac{1}{6}}=\sqrt{3}\\ {\left(x^{\frac{1}{6}}\right)}^6={\left(\sqrt{3}\right)}^6\\ x=27\end{array}$

And,

localid="1643914494055" role="math" $$\begin{gathered} y-3=0 \\ y=3 \\ {x}^{\frac{1}{6}}=3\left[\because y^6=x\Rightarrow y=x^{\frac{1}{6}}\right] \\ {\left(x^{\frac{1}{6}}\right)}^6={\left(3\right)}^6 \\ x=729 \end{gathered}$$

Thus, the real solutions of the given equation are$x=27$and $x=729$.