Calculus Examples

$f (x) = 10 {(x^{2} + y^{2})}^{2}$

Step 1

Rewrite the equation as $10 {(x^{2} + y^{2})}^{2} = x$ .

$10 {(x^{2} + y^{2})}^{2} = x$

Step 2

Divide each term in $10 {(x^{2} + y^{2})}^{2} = x$ by $10$ and simplify.

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Step 2.1

Divide each term in $10 {(x^{2} + y^{2})}^{2} = x$ by $10$ .

$\frac{10 {(x^{2} + y^{2})}^{2}}{10} = \frac{x}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{10 {(x^{2} + y^{2})}^{2}}{10} = \frac{x}{10}$

Step 2.2.1.2

Divide ${(x^{2} + y^{2})}^{2}$ by $1$ .

${(x^{2} + y^{2})}^{2} = \frac{x}{10}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x^{2} + y^{2} = \pm \sqrt{\frac{x}{10}}$

Step 4

Simplify $\pm \sqrt{\frac{x}{10}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{x}{10}}$ as $\frac{\sqrt{x}}{\sqrt{10}}$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x}}{\sqrt{10}}$

Step 4.2

Multiply $\frac{\sqrt{x}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{\sqrt{x}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 4.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{10^{1}}$

Step 4.3.6.5

Evaluate the exponent.

$x^{2} + y^{2} = \pm \frac{\sqrt{x} \sqrt{10}}{10}$

Step 4.4

Combine using the product rule for radicals.

$x^{2} + y^{2} = \pm \frac{\sqrt{x \cdot 10}}{10}$

Step 4.5

Reorder factors in $\pm \frac{\sqrt{x \cdot 10}}{10}$ .

$x^{2} + y^{2} = \pm \frac{\sqrt{10 x}}{10}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x^{2} + y^{2} = \frac{\sqrt{10 x}}{10}$

Step 5.2

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

$y^{2} = \frac{\sqrt{10 x}}{10} - x^{2}$

Step 5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{\sqrt{10 x}}{10} - x^{2}}$

Step 5.4

Simplify $\pm \sqrt{\frac{\sqrt{10 x}}{10} - x^{2}}$ .

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Step 5.4.1

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y = \pm \sqrt{\frac{\sqrt{10 x}}{10} - x^{2} \cdot \frac{10}{10}}$

Step 5.4.2

Combine $- x^{2}$ and $\frac{10}{10}$ .

$y = \pm \sqrt{\frac{\sqrt{10 x}}{10} + \frac{- x^{2} \cdot 10}{10}}$

Step 5.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{\sqrt{10 x} - x^{2} \cdot 10}{10}}$

Step 5.4.4

Multiply $10$ by $- 1$ .

$y = \pm \sqrt{\frac{\sqrt{10 x} - 10 x^{2}}{10}}$

Step 5.4.5

Rewrite $\sqrt{\frac{\sqrt{10 x} - 10 x^{2}}{10}}$ as $\frac{\sqrt{\sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$

Step 5.4.6

Multiply $\frac{\sqrt{\sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 5.4.7

Combine and simplify the denominator.

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Step 5.4.7.1

Multiply $\frac{\sqrt{\sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.4.7.2

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 5.4.7.3

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 5.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 5.4.7.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 5.4.7.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 5.4.7.6.2

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

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 5.4.7.6.3

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

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.4.7.6.4

Cancel the common factor of $2$ .

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Step 5.4.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{1}}$

Step 5.4.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{\sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10}$

Step 5.4.8

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(\sqrt{10 x} - 10 x^{2}) \cdot 10}}{10}$

Step 5.4.9

Reorder factors in $\pm \frac{\sqrt{(\sqrt{10 x} - 10 x^{2}) \cdot 10}}{10}$ .

$y = \pm \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.6

Next, use the negative value of the $\pm$ to find the second solution.

$x^{2} + y^{2} = - \frac{\sqrt{10 x}}{10}$

Step 5.7

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

$y^{2} = - \frac{\sqrt{10 x}}{10} - x^{2}$

Step 5.8

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{\sqrt{10 x}}{10} - x^{2}}$

Step 5.9

Simplify $\pm \sqrt{- \frac{\sqrt{10 x}}{10} - x^{2}}$ .

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Step 5.9.1

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y = \pm \sqrt{- \frac{\sqrt{10 x}}{10} - x^{2} \cdot \frac{10}{10}}$

Step 5.9.2

Combine $- x^{2}$ and $\frac{10}{10}$ .

$y = \pm \sqrt{- \frac{\sqrt{10 x}}{10} + \frac{- x^{2} \cdot 10}{10}}$

Step 5.9.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- \sqrt{10 x} - x^{2} \cdot 10}{10}}$

Step 5.9.4

Multiply $10$ by $- 1$ .

$y = \pm \sqrt{\frac{- \sqrt{10 x} - 10 x^{2}}{10}}$

Step 5.9.5

Rewrite $\sqrt{\frac{- \sqrt{10 x} - 10 x^{2}}{10}}$ as $\frac{\sqrt{- \sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$

Step 5.9.6

Multiply $\frac{\sqrt{- \sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 5.9.7

Combine and simplify the denominator.

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Step 5.9.7.1

Multiply $\frac{\sqrt{- \sqrt{10 x} - 10 x^{2}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.9.7.2

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 5.9.7.3

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 5.9.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 5.9.7.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 5.9.7.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.9.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 5.9.7.6.2

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

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 5.9.7.6.3

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

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.9.7.6.4

Cancel the common factor of $2$ .

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Step 5.9.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.9.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10^{1}}$

Step 5.9.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{- \sqrt{10 x} - 10 x^{2}} \sqrt{10}}{10}$

Step 5.9.8

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(- \sqrt{10 x} - 10 x^{2}) \cdot 10}}{10}$

Step 5.9.9

Reorder factors in $\pm \frac{\sqrt{(- \sqrt{10 x} - 10 x^{2}) \cdot 10}}{10}$ .

$y = \pm \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.10

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.10.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.10.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.10.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

Step 5.11

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (\sqrt{10 x} - 10 x^{2})}}{10}$

$y = \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

$y = - \frac{\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}}{10}$

Step 6

Set the radicand in $\sqrt{10 x}$ greater than or equal to $0$ to find where the expression is defined.

$10 x \geq 0$

Step 7

Divide each term in $10 x \geq 0$ by $10$ and simplify.

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Step 7.1

Divide each term in $10 x \geq 0$ by $10$ .

$\frac{10 x}{10} \geq \frac{0}{10}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $10$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{10 x}{10} \geq \frac{0}{10}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{10}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $10$ .

$x \geq 0$

Step 8

Set the radicand in $\sqrt{10 (\sqrt{10 x} - 10 x^{2})}$ greater than or equal to $0$ to find where the expression is defined.

$10 (\sqrt{10 x} - 10 x^{2}) \geq 0$

Step 9

Solve for $x$ .

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Step 9.1

Divide each term in $10 (\sqrt{10 x} - 10 x^{2}) \geq 0$ by $10$ and simplify.

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Step 9.1.1

Divide each term in $10 (\sqrt{10 x} - 10 x^{2}) \geq 0$ by $10$ .

$\frac{10 (\sqrt{10 x} - 10 x^{2})}{10} \geq \frac{0}{10}$

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $10$ .

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Step 9.1.2.1.1

Cancel the common factor.

$\frac{10 (\sqrt{10 x} - 10 x^{2})}{10} \geq \frac{0}{10}$

Step 9.1.2.1.2

Divide $\sqrt{10 x} - 10 x^{2}$ by $1$ .

$\sqrt{10 x} - 10 x^{2} \geq \frac{0}{10}$

Step 9.1.3

Simplify the right side.

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Step 9.1.3.1

Divide $0$ by $10$ .

$\sqrt{10 x} - 10 x^{2} \geq 0$

Step 9.2

Add $10 x^{2}$ to both sides of the inequality.

$\sqrt{10 x} \geq 10 x^{2}$

Step 9.3

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{10 x}}^{2} \geq {(10 x^{2})}^{2}$

Step 9.4

Simplify each side of the inequality.

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Step 9.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x}$ as ${(10 x)}^{\frac{1}{2}}$ .

${({(10 x)}^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 9.4.2

Simplify the left side.

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Step 9.4.2.1

Simplify ${({(10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 9.4.2.1.1

Multiply the exponents in ${({(10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 9.4.2.1.1.1

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

${(10 x)}^{\frac{1}{2} \cdot 2} \geq {(10 x^{2})}^{2}$

Step 9.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 9.4.2.1.1.2.1

Cancel the common factor.

${(10 x)}^{\frac{1}{2} \cdot 2} \geq {(10 x^{2})}^{2}$

Step 9.4.2.1.1.2.2

Rewrite the expression.

${(10 x)}^{1} \geq {(10 x^{2})}^{2}$

Step 9.4.2.1.2

Simplify.

$10 x \geq {(10 x^{2})}^{2}$

Step 9.4.3

Simplify the right side.

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Step 9.4.3.1

Simplify ${(10 x^{2})}^{2}$ .

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Step 9.4.3.1.1

Apply the product rule to $10 x^{2}$ .

$10 x \geq 10^{2} {(x^{2})}^{2}$

Step 9.4.3.1.2

Raise $10$ to the power of $2$ .

$10 x \geq 100 {(x^{2})}^{2}$

Step 9.4.3.1.3

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

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Step 9.4.3.1.3.1

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

$10 x \geq 100 x^{2 \cdot 2}$

Step 9.4.3.1.3.2

Multiply $2$ by $2$ .

$10 x \geq 100 x^{4}$

Step 9.5

Solve for $x$ .

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Step 9.5.1

Subtract $100 x^{4}$ from both sides of the inequality.

$10 x - 100 x^{4} \geq 0$

Step 9.5.2

Convert the inequality to an equation.

$10 x - 100 x^{4} = 0$

Step 9.5.3

Factor $10 x$ out of $10 x - 100 x^{4}$ .

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Step 9.5.3.1

Factor $10 x$ out of $10 x$ .

$10 x (1) - 100 x^{4} = 0$

Step 9.5.3.2

Factor $10 x$ out of $- 100 x^{4}$ .

$10 x (1) + 10 x (- 10 x^{3}) = 0$

Step 9.5.3.3

Factor $10 x$ out of $10 x (1) + 10 x (- 10 x^{3})$ .

$10 x (1 - 10 x^{3}) = 0$

Step 9.5.4

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

$x = 0$

$1 - 10 x^{3} = 0$

Step 9.5.5

Set $x$ equal to $0$ .

$x = 0$

Step 9.5.6

Set $1 - 10 x^{3}$ equal to $0$ and solve for $x$ .

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Step 9.5.6.1

Set $1 - 10 x^{3}$ equal to $0$ .

$1 - 10 x^{3} = 0$

Step 9.5.6.2

Solve $1 - 10 x^{3} = 0$ for $x$ .

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Step 9.5.6.2.1

Subtract $1$ from both sides of the equation.

$- 10 x^{3} = - 1$

Step 9.5.6.2.2

Divide each term in $- 10 x^{3} = - 1$ by $- 10$ and simplify.

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Step 9.5.6.2.2.1

Divide each term in $- 10 x^{3} = - 1$ by $- 10$ .

$\frac{- 10 x^{3}}{- 10} = \frac{- 1}{- 10}$

Step 9.5.6.2.2.2

Simplify the left side.

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Step 9.5.6.2.2.2.1

Cancel the common factor of $- 10$ .

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Step 9.5.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 10 x^{3}}{- 10} = \frac{- 1}{- 10}$

Step 9.5.6.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{- 1}{- 10}$

Step 9.5.6.2.2.3

Simplify the right side.

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Step 9.5.6.2.2.3.1

Dividing two negative values results in a positive value.

$x^{3} = \frac{1}{10}$

Step 9.5.6.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{1}{10}}$

Step 9.5.6.2.4

Simplify $\sqrt[3]{\frac{1}{10}}$ .

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Step 9.5.6.2.4.1

Rewrite $\sqrt[3]{\frac{1}{10}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{10}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{10}}$

Step 9.5.6.2.4.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{10}}$

Step 9.5.6.2.4.3

Multiply $\frac{1}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$x = \frac{1}{\sqrt[3]{10}} \cdot \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$

Step 9.5.6.2.4.4

Combine and simplify the denominator.

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Step 9.5.6.2.4.4.1

Multiply $\frac{1}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{\sqrt[3]{10} {\sqrt[3]{10}}^{2}}$

Step 9.5.6.2.4.4.2

Raise $\sqrt[3]{10}$ to the power of $1$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1} {\sqrt[3]{10}}^{2}}$

Step 9.5.6.2.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1 + 2}}$

Step 9.5.6.2.4.4.4

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{3}}$

Step 9.5.6.2.4.4.5

Rewrite ${\sqrt[3]{10}}^{3}$ as $10$ .

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Step 9.5.6.2.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10}$ as $10^{\frac{1}{3}}$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{(10^{\frac{1}{3}})}^{3}}$

Step 9.5.6.2.4.4.5.2

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

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{1}{3} \cdot 3}}$

Step 9.5.6.2.4.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 9.5.6.2.4.4.5.4

Cancel the common factor of $3$ .

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Step 9.5.6.2.4.4.5.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 9.5.6.2.4.4.5.4.2

Rewrite the expression.

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{1}}$

Step 9.5.6.2.4.4.5.5

Evaluate the exponent.

$x = \frac{{\sqrt[3]{10}}^{2}}{10}$

Step 9.5.6.2.4.5

Simplify the numerator.

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Step 9.5.6.2.4.5.1

Rewrite ${\sqrt[3]{10}}^{2}$ as $\sqrt[3]{10^{2}}$ .

$x = \frac{\sqrt[3]{10^{2}}}{10}$

Step 9.5.6.2.4.5.2

Raise $10$ to the power of $2$ .

$x = \frac{\sqrt[3]{100}}{10}$

Step 9.5.7

The final solution is all the values that make $10 x (1 - 10 x^{3}) = 0$ true.

$x = 0, \frac{\sqrt[3]{100}}{10}$

Step 9.6

Find the domain of $10 (\sqrt{10 x} - 10 x^{2})$ .

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Step 9.6.1

Set the radicand in $\sqrt{10 x}$ greater than or equal to $0$ to find where the expression is defined.

$10 x \geq 0$

Step 9.6.2

Divide each term in $10 x \geq 0$ by $10$ and simplify.

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Step 9.6.2.1

Divide each term in $10 x \geq 0$ by $10$ .

$\frac{10 x}{10} \geq \frac{0}{10}$

Step 9.6.2.2

Simplify the left side.

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Step 9.6.2.2.1

Cancel the common factor of $10$ .

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Step 9.6.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} \geq \frac{0}{10}$

Step 9.6.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{10}$

Step 9.6.2.3

Simplify the right side.

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Step 9.6.2.3.1

Divide $0$ by $10$ .

$x \geq 0$

Step 9.6.3

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 9.7

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{\sqrt[3]{100}}{10}$

$x > \frac{\sqrt[3]{100}}{10}$

Step 9.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 9.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 9.8.1.2

Replace $x$ with $- 2$ in the original inequality.

$10 (\sqrt{10 (- 2)} - 10 {(- 2)}^{2}) \geq 0$

Step 9.8.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 9.8.2

Test a value on the interval $0 < x < \frac{\sqrt[3]{100}}{10}$ to see if it makes the inequality true.

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Step 9.8.2.1

Choose a value on the interval $0 < x < \frac{\sqrt[3]{100}}{10}$ and see if this value makes the original inequality true.

$x = 0.23$

Step 9.8.2.2

Replace $x$ with $0.23$ in the original inequality.

$10 (\sqrt{10 (0.23)} - 10 {(0.23)}^{2}) \geq 0$

Step 9.8.2.3

The left side $9.87575088$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.8.3

Test a value on the interval $x > \frac{\sqrt[3]{100}}{10}$ to see if it makes the inequality true.

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Step 9.8.3.1

Choose a value on the interval $x > \frac{\sqrt[3]{100}}{10}$ and see if this value makes the original inequality true.

$x = 3$

Step 9.8.3.2

Replace $x$ with $3$ in the original inequality.

$10 (\sqrt{10 (3)} - 10 {(3)}^{2}) \geq 0$

Step 9.8.3.3

The left side $- 845.22774424$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < \frac{\sqrt[3]{100}}{10}$ True

$x > \frac{\sqrt[3]{100}}{10}$ False

$x < 0$ False

$0 < x < \frac{\sqrt[3]{100}}{10}$ True

$x > \frac{\sqrt[3]{100}}{10}$ False

Step 9.9

The solution consists of all of the true intervals.

$0 \leq x \leq \frac{\sqrt[3]{100}}{10}$

Step 10

Set the radicand in $\sqrt{10 (- \sqrt{10 x} - 10 x^{2})}$ greater than or equal to $0$ to find where the expression is defined.

$10 (- \sqrt{10 x} - 10 x^{2}) \geq 0$

Step 11

Solve for $x$ .

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Step 11.1

Divide each term in $10 (- \sqrt{10 x} - 10 x^{2}) \geq 0$ by $10$ and simplify.

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Step 11.1.1

Divide each term in $10 (- \sqrt{10 x} - 10 x^{2}) \geq 0$ by $10$ .

$\frac{10 (- \sqrt{10 x} - 10 x^{2})}{10} \geq \frac{0}{10}$

Step 11.1.2

Simplify the left side.

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Step 11.1.2.1

Cancel the common factor of $10$ .

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Step 11.1.2.1.1

Cancel the common factor.

$\frac{10 (- \sqrt{10 x} - 10 x^{2})}{10} \geq \frac{0}{10}$

Step 11.1.2.1.2

Divide $- \sqrt{10 x} - 10 x^{2}$ by $1$ .

$- \sqrt{10 x} - 10 x^{2} \geq \frac{0}{10}$

Step 11.1.3

Simplify the right side.

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Step 11.1.3.1

Divide $0$ by $10$ .

$- \sqrt{10 x} - 10 x^{2} \geq 0$

Step 11.2

Add $10 x^{2}$ to both sides of the inequality.

$- \sqrt{10 x} \geq 10 x^{2}$

Step 11.3

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(- \sqrt{10 x})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4

Simplify each side of the inequality.

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Step 11.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x}$ as ${(10 x)}^{\frac{1}{2}}$ .

${(- {(10 x)}^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2

Simplify the left side.

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Step 11.4.2.1

Simplify ${(- {(10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 11.4.2.1.1

Apply the product rule to $10 x$ .

${(- (10^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.4.2.1.2.1

Apply the product rule to $- 10^{\frac{1}{2}} x^{\frac{1}{2}}$ .

${(- 10^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.2.2

Apply the product rule to $- 10^{\frac{1}{2}}$ .

${(- 1)}^{2} {(10^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.3

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

$1 {(10^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.4

Multiply ${(10^{\frac{1}{2}})}^{2}$ by $1$ .

${(10^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.5

Multiply the exponents in ${(10^{\frac{1}{2}})}^{2}$ .

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Step 11.4.2.1.5.1

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

$10^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.5.2

Cancel the common factor of $2$ .

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Step 11.4.2.1.5.2.1

Cancel the common factor.

$10^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.5.2.2

Rewrite the expression.

$10^{1} {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.6

Evaluate the exponent.

$10 {(x^{\frac{1}{2}})}^{2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.7

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

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Step 11.4.2.1.7.1

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

$10 x^{\frac{1}{2} \cdot 2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.7.2

Cancel the common factor of $2$ .

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Step 11.4.2.1.7.2.1

Cancel the common factor.

$10 x^{\frac{1}{2} \cdot 2} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.7.2.2

Rewrite the expression.

$10 x^{1} \geq {(10 x^{2})}^{2}$

Step 11.4.2.1.8

Simplify.

$10 x \geq {(10 x^{2})}^{2}$

Step 11.4.3

Simplify the right side.

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Step 11.4.3.1

Simplify ${(10 x^{2})}^{2}$ .

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Step 11.4.3.1.1

Apply the product rule to $10 x^{2}$ .

$10 x \geq 10^{2} {(x^{2})}^{2}$

Step 11.4.3.1.2

Raise $10$ to the power of $2$ .

$10 x \geq 100 {(x^{2})}^{2}$

Step 11.4.3.1.3

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

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Step 11.4.3.1.3.1

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

$10 x \geq 100 x^{2 \cdot 2}$

Step 11.4.3.1.3.2

Multiply $2$ by $2$ .

$10 x \geq 100 x^{4}$

Step 11.5

Solve for $x$ .

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Step 11.5.1

Subtract $100 x^{4}$ from both sides of the inequality.

$10 x - 100 x^{4} \geq 0$

Step 11.5.2

Convert the inequality to an equation.

$10 x - 100 x^{4} = 0$

Step 11.5.3

Factor $10 x$ out of $10 x - 100 x^{4}$ .

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Step 11.5.3.1

Factor $10 x$ out of $10 x$ .

$10 x (1) - 100 x^{4} = 0$

Step 11.5.3.2

Factor $10 x$ out of $- 100 x^{4}$ .

$10 x (1) + 10 x (- 10 x^{3}) = 0$

Step 11.5.3.3

Factor $10 x$ out of $10 x (1) + 10 x (- 10 x^{3})$ .

$10 x (1 - 10 x^{3}) = 0$

Step 11.5.4

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

$x = 0$

$1 - 10 x^{3} = 0$

Step 11.5.5

Set $x$ equal to $0$ .

$x = 0$

Step 11.5.6

Set $1 - 10 x^{3}$ equal to $0$ and solve for $x$ .

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Step 11.5.6.1

Set $1 - 10 x^{3}$ equal to $0$ .

$1 - 10 x^{3} = 0$

Step 11.5.6.2

Solve $1 - 10 x^{3} = 0$ for $x$ .

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Step 11.5.6.2.1

Subtract $1$ from both sides of the equation.

$- 10 x^{3} = - 1$

Step 11.5.6.2.2

Divide each term in $- 10 x^{3} = - 1$ by $- 10$ and simplify.

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Step 11.5.6.2.2.1

Divide each term in $- 10 x^{3} = - 1$ by $- 10$ .

$\frac{- 10 x^{3}}{- 10} = \frac{- 1}{- 10}$

Step 11.5.6.2.2.2

Simplify the left side.

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Step 11.5.6.2.2.2.1

Cancel the common factor of $- 10$ .

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Step 11.5.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 10 x^{3}}{- 10} = \frac{- 1}{- 10}$

Step 11.5.6.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{- 1}{- 10}$

Step 11.5.6.2.2.3

Simplify the right side.

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Step 11.5.6.2.2.3.1

Dividing two negative values results in a positive value.

$x^{3} = \frac{1}{10}$

Step 11.5.6.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{1}{10}}$

Step 11.5.6.2.4

Simplify $\sqrt[3]{\frac{1}{10}}$ .

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Step 11.5.6.2.4.1

Rewrite $\sqrt[3]{\frac{1}{10}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{10}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{10}}$

Step 11.5.6.2.4.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{10}}$

Step 11.5.6.2.4.3

Multiply $\frac{1}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$x = \frac{1}{\sqrt[3]{10}} \cdot \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$

Step 11.5.6.2.4.4

Combine and simplify the denominator.

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Step 11.5.6.2.4.4.1

Multiply $\frac{1}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{\sqrt[3]{10} {\sqrt[3]{10}}^{2}}$

Step 11.5.6.2.4.4.2

Raise $\sqrt[3]{10}$ to the power of $1$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1} {\sqrt[3]{10}}^{2}}$

Step 11.5.6.2.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1 + 2}}$

Step 11.5.6.2.4.4.4

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{3}}$

Step 11.5.6.2.4.4.5

Rewrite ${\sqrt[3]{10}}^{3}$ as $10$ .

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Step 11.5.6.2.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10}$ as $10^{\frac{1}{3}}$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{{(10^{\frac{1}{3}})}^{3}}$

Step 11.5.6.2.4.4.5.2

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

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{1}{3} \cdot 3}}$

Step 11.5.6.2.4.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 11.5.6.2.4.4.5.4

Cancel the common factor of $3$ .

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Step 11.5.6.2.4.4.5.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 11.5.6.2.4.4.5.4.2

Rewrite the expression.

$x = \frac{{\sqrt[3]{10}}^{2}}{10^{1}}$

Step 11.5.6.2.4.4.5.5

Evaluate the exponent.

$x = \frac{{\sqrt[3]{10}}^{2}}{10}$

Step 11.5.6.2.4.5

Simplify the numerator.

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Step 11.5.6.2.4.5.1

Rewrite ${\sqrt[3]{10}}^{2}$ as $\sqrt[3]{10^{2}}$ .

$x = \frac{\sqrt[3]{10^{2}}}{10}$

Step 11.5.6.2.4.5.2

Raise $10$ to the power of $2$ .

$x = \frac{\sqrt[3]{100}}{10}$

Step 11.5.7

The final solution is all the values that make $10 x (1 - 10 x^{3}) = 0$ true.

$x = 0, \frac{\sqrt[3]{100}}{10}$

Step 12

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \frac{\sqrt[3]{100}}{10}]$

Set-Builder Notation:

${x | 0 \leq x \leq \frac{\sqrt[3]{100}}{10}}$

Step 13

The range is the set of all valid $y$ values. Use the graph to find the range.

No solution

Step 14

Determine the domain and range.

No solution

Step 15