Solve for x square root of x=x^3

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

Step 1

Subtract $x^{3}$ from both sides of the equation.

$\sqrt{x} - x^{3} = 0$

Step 2

Factor $\sqrt{x} - x^{3}$ .

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

$x^{\frac{1}{2}} - x^{3} = 0$

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} - x^{3}$ .

Multiply by $1$ .

$x^{\frac{1}{2}} \cdot 1 - x^{3} = 0$

Factor $x^{\frac{1}{2}}$ out of $- x^{3}$ .

$x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{5}{2}}) = 0$

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{5}{2}})$ .

$x^{\frac{1}{2}} (1 - x^{\frac{5}{2}}) = 0$

Step 3

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

$x^{\frac{1}{2}} = 0$

$1 - x^{\frac{5}{2}} = 0$

Step 4

Set $x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}}$ equal to $0$ .

$x^{\frac{1}{2}} = 0$

Solve $x^{\frac{1}{2}} = 0$ for $x$ .

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

$x^{1} = 0^{2}$

Simplify.

$x = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 5

Set $1 - x^{\frac{5}{2}}$ equal to $0$ and solve for $x$ .

Set $1 - x^{\frac{5}{2}}$ equal to $0$ .

$1 - x^{\frac{5}{2}} = 0$

Solve $1 - x^{\frac{5}{2}} = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x^{\frac{5}{2}} = - 1$

Raise each side of the equation to the power of $\frac{2}{5}$ to eliminate the fractional exponent on the left side.

${(- x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(- x^{\frac{5}{2}})}^{\frac{2}{5}}$ .

Simplify the expression.

Apply the product rule to $- x^{\frac{5}{2}}$ .

${(- 1)}^{\frac{2}{5}} {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Rewrite $- 1$ as ${(- 1)}^{5}$ .

${({(- 1)}^{5})}^{\frac{2}{5}} {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

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

${(- 1)}^{5 (\frac{2}{5})} {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(- 1)}^{5 (\frac{2}{5})} {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Rewrite the expression.

${(- 1)}^{2} {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 {(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Multiply ${(x^{\frac{5}{2}})}^{\frac{2}{5}}$ by $1$ .

${(x^{\frac{5}{2}})}^{\frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Multiply the exponents in ${(x^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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

$x^{\frac{5}{2} \cdot \frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{5}{2} \cdot \frac{2}{5}} = {(- 1)}^{\frac{2}{5}}$

Rewrite the expression.

$x^{\frac{1}{2} \cdot 2} = {(- 1)}^{\frac{2}{5}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- 1)}^{\frac{2}{5}}$

Rewrite the expression.

$x^{1} = {(- 1)}^{\frac{2}{5}}$

Simplify.

$x = {(- 1)}^{\frac{2}{5}}$

Simplify the right side.

Simplify ${(- 1)}^{\frac{2}{5}}$ .

Simplify the expression.

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$x = {({(- 1)}^{5})}^{\frac{2}{5}}$

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

$x = {(- 1)}^{5 (\frac{2}{5})}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x = {(- 1)}^{5 (\frac{2}{5})}$

Rewrite the expression.

$x = {(- 1)}^{2}$

Raise $- 1$ to the power of $2$ .

$x = 1$

Step 6

The final solution is all the values that make $x^{\frac{1}{2}} (1 - x^{\frac{5}{2}}) = 0$ true.

$x = 0, 1$

Solve for x square root of x=x^3

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