Calculus Examples

$x = \sqrt[6]{x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt[6]{x} = x$

Step 2

Subtract $x$ from both sides of the equation.

$\sqrt[6]{x} - x = 0$

Step 3

Factor $\sqrt[6]{x} - x$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$x^{\frac{1}{6}} - x = 0$

Step 3.2

Factor $x^{\frac{1}{6}}$ out of $x^{\frac{1}{6}} - x$ .

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Step 3.2.1

Multiply by $1$ .

$x^{\frac{1}{6}} \cdot 1 - x = 0$

Step 3.2.2

Factor $x^{\frac{1}{6}}$ out of $- x$ .

$x^{\frac{1}{6}} \cdot 1 + x^{\frac{1}{6}} (- x^{\frac{5}{6}}) = 0$

Step 3.2.3

Factor $x^{\frac{1}{6}}$ out of $x^{\frac{1}{6}} \cdot 1 + x^{\frac{1}{6}} (- x^{\frac{5}{6}})$ .

$x^{\frac{1}{6}} (1 - x^{\frac{5}{6}}) = 0$

Step 4

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

$x^{\frac{1}{6}} = 0$

$1 - x^{\frac{5}{6}} = 0$

Step 5

Set $x^{\frac{1}{6}}$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{\frac{1}{6}}$ equal to $0$ .

$x^{\frac{1}{6}} = 0$

Step 5.2

Solve $x^{\frac{1}{6}} = 0$ for $x$ .

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Step 5.2.1

Raise each side of the equation to the power of $6$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{6}})}^{6} = 0^{6}$

Step 5.2.2

Simplify the exponent.

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Step 5.2.2.1

Simplify the left side.

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Step 5.2.2.1.1

Simplify ${(x^{\frac{1}{6}})}^{6}$ .

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Step 5.2.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{6}$ .

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Step 5.2.2.1.1.1.1

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

$x^{\frac{1}{6} \cdot 6} = 0^{6}$

Step 5.2.2.1.1.1.2

Cancel the common factor of $6$ .

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Step 5.2.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{6} \cdot 6} = 0^{6}$

Step 5.2.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{6}$

Step 5.2.2.1.1.2

Simplify.

$x = 0^{6}$

Step 5.2.2.2

Simplify the right side.

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Step 5.2.2.2.1

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

$x = 0$

Step 6

Set $1 - x^{\frac{5}{6}}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $1 - x^{\frac{5}{6}}$ equal to $0$ .

$1 - x^{\frac{5}{6}} = 0$

Step 6.2

Solve $1 - x^{\frac{5}{6}} = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$- x^{\frac{5}{6}} = - 1$

Step 6.2.2

Raise each side of the equation to the power of $\frac{6}{5}$ to eliminate the fractional exponent on the left side.

${(- x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify ${(- x^{\frac{5}{6}})}^{\frac{6}{5}}$ .

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Step 6.2.3.1.1.1

Simplify the expression.

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Step 6.2.3.1.1.1.1

Apply the product rule to $- x^{\frac{5}{6}}$ .

${(- 1)}^{\frac{6}{5}} {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.1.2

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

${({(- 1)}^{5})}^{\frac{6}{5}} {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.1.3

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

${(- 1)}^{5 (\frac{6}{5})} {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.2

Cancel the common factor of $5$ .

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Step 6.2.3.1.1.2.1

Cancel the common factor.

${(- 1)}^{5 (\frac{6}{5})} {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.2.2

Rewrite the expression.

${(- 1)}^{6} {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3

Simplify the expression.

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Step 6.2.3.1.1.3.1

Raise $- 1$ to the power of $6$ .

$1 {(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.2

Multiply ${(x^{\frac{5}{6}})}^{\frac{6}{5}}$ by $1$ .

${(x^{\frac{5}{6}})}^{\frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.3

Multiply the exponents in ${(x^{\frac{5}{6}})}^{\frac{6}{5}}$ .

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Step 6.2.3.1.1.3.3.1

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

$x^{\frac{5}{6} \cdot \frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.3.2

Cancel the common factor of $5$ .

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Step 6.2.3.1.1.3.3.2.1

Cancel the common factor.

$x^{\frac{5}{6} \cdot \frac{6}{5}} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.3.2.2

Rewrite the expression.

$x^{\frac{1}{6} \cdot 6} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.3.3

Cancel the common factor of $6$ .

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Step 6.2.3.1.1.3.3.3.1

Cancel the common factor.

$x^{\frac{1}{6} \cdot 6} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.3.3.3.2

Rewrite the expression.

$x^{1} = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.1.1.4

Simplify.

$x = {(- 1)}^{\frac{6}{5}}$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Simplify ${(- 1)}^{\frac{6}{5}}$ .

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Step 6.2.3.2.1.1

Simplify the expression.

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Step 6.2.3.2.1.1.1

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

$x = {({(- 1)}^{5})}^{\frac{6}{5}}$

Step 6.2.3.2.1.1.2

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

$x = {(- 1)}^{5 (\frac{6}{5})}$

Step 6.2.3.2.1.2

Cancel the common factor of $5$ .

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Step 6.2.3.2.1.2.1

Cancel the common factor.

$x = {(- 1)}^{5 (\frac{6}{5})}$

Step 6.2.3.2.1.2.2

Rewrite the expression.

$x = {(- 1)}^{6}$

Step 6.2.3.2.1.3

Raise $- 1$ to the power of $6$ .

$x = 1$

Step 7

The final solution is all the values that make $x^{\frac{1}{6}} (1 - x^{\frac{5}{6}}) = 0$ true.

$x = 0, 1$