Pre-Algebra Examples

$x {(x^{2} - 4 x)}^{\frac{1}{3}} + 2 {(x^{2} - 4 x)}^{\frac{4}{3}} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor ${(x^{2} - 4 x)}^{\frac{1}{3}}$ out of $x {(x^{2} - 4 x)}^{\frac{1}{3}} + 2 {(x^{2} - 4 x)}^{\frac{4}{3}}$ .

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Step 1.1.1

Factor ${(x^{2} - 4 x)}^{\frac{1}{3}}$ out of $x {(x^{2} - 4 x)}^{\frac{1}{3}}$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} x + 2 {(x^{2} - 4 x)}^{\frac{4}{3}} = 0$

Step 1.1.2

Factor ${(x^{2} - 4 x)}^{\frac{1}{3}}$ out of $2 {(x^{2} - 4 x)}^{\frac{4}{3}}$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} x + {(x^{2} - 4 x)}^{\frac{1}{3}} (2 {(x^{2} - 4 x)}^{\frac{3}{3}}) = 0$

Step 1.1.3

Factor ${(x^{2} - 4 x)}^{\frac{1}{3}}$ out of ${(x^{2} - 4 x)}^{\frac{1}{3}} x + {(x^{2} - 4 x)}^{\frac{1}{3}} (2 {(x^{2} - 4 x)}^{\frac{3}{3}})$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (x + 2 {(x^{2} - 4 x)}^{\frac{3}{3}}) = 0$

Step 1.2

Divide $3$ by $3$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (x + 2 (x^{2} - 4 x)) = 0$

Step 1.3

Simplify.

${(x^{2} - 4 x)}^{\frac{1}{3}} (x + 2 (x^{2} - 4 x)) = 0$

Step 1.4

Apply the distributive property.

${(x^{2} - 4 x)}^{\frac{1}{3}} (x + 2 x^{2} + 2 (- 4 x)) = 0$

Step 1.5

Multiply $- 4$ by $2$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (x + 2 x^{2} - 8 x) = 0$

Step 1.6

Subtract $8 x$ from $x$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (2 x^{2} - 7 x) = 0$

Step 1.7

Factor.

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Step 1.7.1

Factor $x$ out of $2 x^{2} - 7 x$ .

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Step 1.7.1.1

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

${(x^{2} - 4 x)}^{\frac{1}{3}} (x (2 x) - 7 x) = 0$

Step 1.7.1.2

Factor $x$ out of $- 7 x$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (x (2 x) + x \cdot - 7) = 0$

Step 1.7.1.3

Factor $x$ out of $x (2 x) + x \cdot - 7$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} (x (2 x - 7)) = 0$

Step 1.7.2

Remove unnecessary parentheses.

${(x^{2} - 4 x)}^{\frac{1}{3}} x (2 x - 7) = 0$

Step 2

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

${(x^{2} - 4 x)}^{\frac{1}{3}} = 0$

$x = 0$

$2 x - 7 = 0$

Step 3

Set ${(x^{2} - 4 x)}^{\frac{1}{3}}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set ${(x^{2} - 4 x)}^{\frac{1}{3}}$ equal to $0$ .

${(x^{2} - 4 x)}^{\frac{1}{3}} = 0$

Step 3.2

Solve ${(x^{2} - 4 x)}^{\frac{1}{3}} = 0$ for $x$ .

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Step 3.2.1

Set the $x^{2} - 4 x$ equal to $0$ .

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

Step 3.2.2

Solve for $x$ .

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

$x \cdot x - 4 x = 0$

Step 3.2.2.1.2

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Step 3.2.2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

Step 3.2.2.2

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

$x = 0$

$x - 4 = 0$

Step 3.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.2.2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 3.2.2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.2.2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.2.2.5

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 5.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 5.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 5.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 6

The final solution is all the values that make ${(x^{2} - 4 x)}^{\frac{1}{3}} x (2 x - 7) = 0$ true.

$x = 0, 4, \frac{7}{2}$