Solve sqrt[3]{9}x=sqrt{3x}

$$\sqrt[3]{ 9 } x= \sqrt{ 3x }$$

Solve for x

$$x=\frac{3^{\frac{2}{3}}}{3}\approx 0.693361274$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt[3]{9}x\right)^{2}=\left(\sqrt{3x}\right)^{2}$$

Expand \left(\sqrt[3]{9}x\right)^{2}.

$$\left(\sqrt[3]{9}\right)^{2}x^{2}=\left(\sqrt{3x}\right)^{2}$$

Calculate \sqrt{3x} to the power of 2 and get 3x.

$$\left(\sqrt[3]{9}\right)^{2}x^{2}=3x$$

Subtract 3x from both sides.

$$\left(\sqrt[3]{9}\right)^{2}x^{2}-3x=0$$

Factor out x.

$$x\left(\left(\sqrt[3]{9}\right)^{2}x-3\right)=0$$

To find equation solutions, solve x=0 and \left(\sqrt[3]{9}\right)^{2}x-3=0.

$$x=0 x=\frac{3}{\left(\sqrt[3]{9}\right)^{2}}$$

Substitute 0 for x in the equation \sqrt[3]{9}x=\sqrt{3x}.

$$\sqrt[3]{9}\times 0=\sqrt{3\times 0}$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Substitute \frac{3}{\left(\sqrt[3]{9}\right)^{2}} for x in the equation \sqrt[3]{9}x=\sqrt{3x}.

$$\sqrt[3]{9}\times \frac{3}{\left(\sqrt[3]{9}\right)^{2}}=\sqrt{3\times \frac{3}{\left(\sqrt[3]{9}\right)^{2}}}$$

Simplify. The value x=\frac{3}{\left(\sqrt[3]{9}\right)^{2}} satisfies the equation.

$$3\times 9^{-\frac{1}{3}}=3\times 9^{-\frac{1}{3}}$$

List all solutions of \sqrt[3]{9}x=\sqrt{3x}.

$$x=0 x=\frac{3}{\left(\sqrt[3]{9}\right)^{2}}$$