Pre-Algebra Examples

$\sqrt{x} - \sqrt[3]{x (x - 1)} < 0$

Step 1

Add $\sqrt[3]{x (x - 1)}$ to both sides of the inequality.

$\sqrt{x} < \sqrt[3]{x (x - 1)}$

Step 2

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

${\sqrt{x}}^{2} < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

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

${(x^{\frac{1}{2}})}^{2} < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3.2.1.2

Simplify.

$x < {\sqrt[3]{x (x - 1)}}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${\sqrt[3]{x (x - 1)}}^{2}$ .

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Step 3.3.1.1

Rewrite ${\sqrt[3]{x (x - 1)}}^{2}$ as $\sqrt[3]{{(x (x - 1))}^{2}}$ .

$x < \sqrt[3]{{(x (x - 1))}^{2}}$

Step 3.3.1.2

Apply the product rule to $x (x - 1)$ .

$x < \sqrt[3]{x^{2} {(x - 1)}^{2}}$

Step 4

Rewrite so $\sqrt[3]{x^{2} {(x - 1)}^{2}}$ is on the left side of the inequality.

$\sqrt[3]{x^{2} {(x - 1)}^{2}} > x$

Step 5

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

${\sqrt[3]{x^{2} {(x - 1)}^{2}}}^{3} > x^{3}$

Step 6

Simplify each side of the inequality.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} {(x - 1)}^{2}}$ as ${(x^{2} {(x - 1)}^{2})}^{\frac{1}{3}}$ .

${({(x^{2} {(x - 1)}^{2})}^{\frac{1}{3}})}^{3} > x^{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${({(x^{2} {(x - 1)}^{2})}^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1

Multiply the exponents in ${({(x^{2} {(x - 1)}^{2})}^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.1.2.1

Cancel the common factor.

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

Step 6.2.1.1.2.2

Rewrite the expression.

${(x^{2} {(x - 1)}^{2})}^{1} > x^{3}$

Step 6.2.1.2

Simplify.

$x^{2} {(x - 1)}^{2} > x^{3}$

Step 7

Solve for $x$ .

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Step 7.1

Move all terms containing $x$ to the left side of the inequality.

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Step 7.1.1

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

$x^{2} {(x - 1)}^{2} - x^{3} > 0$

Step 7.1.2

Simplify each term.

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Step 7.1.2.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x^{2} ((x - 1) (x - 1)) - x^{3} > 0$

Step 7.1.2.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 7.1.2.2.1

Apply the distributive property.

$x^{2} (x (x - 1) - 1 (x - 1)) - x^{3} > 0$

Step 7.1.2.2.2

Apply the distributive property.

$x^{2} (x \cdot x + x \cdot - 1 - 1 (x - 1)) - x^{3} > 0$

Step 7.1.2.2.3

Apply the distributive property.

$x^{2} (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - x^{3} > 0$

Step 7.1.2.3

Simplify and combine like terms.

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Step 7.1.2.3.1

Simplify each term.

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Step 7.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - x^{3} > 0$

Step 7.1.2.3.1.2

Move $- 1$ to the left of $x$ .

$x^{2} (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - x^{3} > 0$

Step 7.1.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} (x^{2} - x - 1 x - 1 \cdot - 1) - x^{3} > 0$

Step 7.1.2.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} (x^{2} - x - x - 1 \cdot - 1) - x^{3} > 0$

Step 7.1.2.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} (x^{2} - x - x + 1) - x^{3} > 0$

Step 7.1.2.3.2

Subtract $x$ from $- x$ .

$x^{2} (x^{2} - 2 x + 1) - x^{3} > 0$

Step 7.1.2.4

Apply the distributive property.

$x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 - x^{3} > 0$

Step 7.1.2.5

Simplify.

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Step 7.1.2.5.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 7.1.2.5.1.1

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

$x^{2 + 2} + x^{2} (- 2 x) + x^{2} \cdot 1 - x^{3} > 0$

Step 7.1.2.5.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 - x^{3} > 0$

Step 7.1.2.5.2

Rewrite using the commutative property of multiplication.

$x^{4} - 2 x^{2} x + x^{2} \cdot 1 - x^{3} > 0$

Step 7.1.2.5.3

Multiply $x^{2}$ by $1$ .

$x^{4} - 2 x^{2} x + x^{2} - x^{3} > 0$

Step 7.1.2.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.1.2.6.1

Move $x$ .

$x^{4} - 2 (x \cdot x^{2}) + x^{2} - x^{3} > 0$

Step 7.1.2.6.2

Multiply $x$ by $x^{2}$ .

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Step 7.1.2.6.2.1

Raise $x$ to the power of $1$ .

$x^{4} - 2 (x^{1} x^{2}) + x^{2} - x^{3} > 0$

Step 7.1.2.6.2.2

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

$x^{4} - 2 x^{1 + 2} + x^{2} - x^{3} > 0$

Step 7.1.2.6.3

Add $1$ and $2$ .

$x^{4} - 2 x^{3} + x^{2} - x^{3} > 0$

Step 7.1.3

Subtract $x^{3}$ from $- 2 x^{3}$ .

$x^{4} - 3 x^{3} + x^{2} > 0$

Step 7.2

Convert the inequality to an equation.

$x^{4} - 3 x^{3} + x^{2} = 0$

Step 7.3

Factor $x^{2}$ out of $x^{4} - 3 x^{3} + x^{2}$ .

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Step 7.3.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 7.3.2

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

$x^{2} x^{2} + x^{2} (- 3 x) + x^{2} = 0$

Step 7.3.3

Multiply by $1$ .

$x^{2} x^{2} + x^{2} (- 3 x) + x^{2} \cdot 1 = 0$

Step 7.3.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 3 x)$ .

$x^{2} (x^{2} - 3 x) + x^{2} \cdot 1 = 0$

Step 7.3.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 3 x) + x^{2} \cdot 1$ .

$x^{2} (x^{2} - 3 x + 1) = 0$

Step 7.4

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

$x^{2} = 0$

$x^{2} - 3 x + 1 = 0$

Step 7.5

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

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Step 7.5.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 7.5.2

Solve $x^{2} = 0$ for $x$ .

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Step 7.5.2.1

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

$x = \pm \sqrt{0}$

Step 7.5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.5.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 7.5.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 7.5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7.6

Set $x^{2} - 3 x + 1$ equal to $0$ and solve for $x$ .

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Step 7.6.1

Set $x^{2} - 3 x + 1$ equal to $0$ .

$x^{2} - 3 x + 1 = 0$

Step 7.6.2

Solve $x^{2} - 3 x + 1 = 0$ for $x$ .

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Step 7.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.6.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 7.6.2.3

Simplify.

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Step 7.6.2.3.1

Simplify the numerator.

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Step 7.6.2.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 7.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 7.6.2.3.1.3

Subtract $4$ from $9$ .

$x = \frac{3 \pm \sqrt{5}}{2 \cdot 1}$

Step 7.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 7.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.6.2.4.1

Simplify the numerator.

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Step 7.6.2.4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 7.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 7.6.2.4.1.3

Subtract $4$ from $9$ .

$x = \frac{3 \pm \sqrt{5}}{2 \cdot 1}$

Step 7.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 7.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{5}}{2}$

Step 7.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.6.2.5.1

Simplify the numerator.

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Step 7.6.2.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 7.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 7.6.2.5.1.3

Subtract $4$ from $9$ .

$x = \frac{3 \pm \sqrt{5}}{2 \cdot 1}$

Step 7.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 7.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{5}}{2}$

Step 7.6.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 7.7

The final solution is all the values that make $x^{2} (x^{2} - 3 x + 1) = 0$ true.

$x = 0, \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 8

Find the domain of $\sqrt{x} - \sqrt[3]{x (x - 1)}$ .

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Step 8.1

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

$x \geq 0$

Step 8.2

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

$[0, \infty)$

Step 9

The solution consists of all of the true intervals.

$x > \frac{3 + \sqrt{5}}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{3 + \sqrt{5}}{2}$

Interval Notation:

$(\frac{3 + \sqrt{5}}{2}, \infty)$

Step 11