Solve for x fourth root of 3x = square root of x

$\sqrt[4]{3 x} = \sqrt{x}$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{3 x}}^{4} = {\sqrt{x}}^{4}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 x}$ as ${(3 x)}^{\frac{1}{4}}$ .

${({(3 x)}^{\frac{1}{4}})}^{4} = {\sqrt{x}}^{4}$

Simplify the left side.

Simplify ${({(3 x)}^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${({(3 x)}^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 x)}^{\frac{1}{4} \cdot 4} = {\sqrt{x}}^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(3 x)}^{\frac{1}{4} \cdot 4} = {\sqrt{x}}^{4}$

Rewrite the expression.

${(3 x)}^{1} = {\sqrt{x}}^{4}$

Simplify.

$3 x = {\sqrt{x}}^{4}$

Simplify the right side.

Rewrite ${\sqrt{x}}^{4}$ as $x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$3 x = {(x^{\frac{1}{2}})}^{4}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 x = x^{\frac{1}{2} \cdot 4}$

Combine $\frac{1}{2}$ and $4$ .

$3 x = x^{\frac{4}{2}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$3 x = x^{\frac{2 \cdot 2}{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$3 x = x^{\frac{2 \cdot 2}{2 (1)}}$

Cancel the common factor.

$3 x = x^{\frac{2 \cdot 2}{2 \cdot 1}}$

Rewrite the expression.

$3 x = x^{\frac{2}{1}}$

Divide $2$ by $1$ .

$3 x = x^{2}$

Step 3

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

$3 x - x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$3 u - u^{2} = 0$

Factor $u$ out of $3 u - u^{2}$ .

Factor $u$ out of $3 u$ .

$u \cdot 3 - u^{2} = 0$

Factor $u$ out of $- u^{2}$ .

$u \cdot 3 + u (- u) = 0$

Factor $u$ out of $u \cdot 3 + u (- u)$ .

$u (3 - u) = 0$

Replace all occurrences of $u$ with $x$ .

$x (3 - x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $3 - x$ equal to $0$ and solve for $x$ .

Set $3 - x$ equal to $0$ .

$3 - x = 0$

Solve $3 - x = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

The final solution is all the values that make $x (3 - x) = 0$ true.

$x = 0, 3$

Solve for x fourth root of 3x = square root of x

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