Basic Math Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Factor ${(y^{2} - 4)}^{\frac{1}{3}}$ out of $8 y^{2} {(y^{2} - 4)}^{\frac{1}{3}} + {(y^{2} - 4)}^{\frac{4}{3}}$ .

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Step 1.1.1

Factor ${(y^{2} - 4)}^{\frac{1}{3}}$ out of $8 y^{2} {(y^{2} - 4)}^{\frac{1}{3}}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor ${(y^{2} - 4)}^{\frac{1}{3}}$ out of ${(y^{2} - 4)}^{\frac{1}{3}} (8 y^{2}) + {(y^{2} - 4)}^{\frac{1}{3}} {(y^{2} - 4)}^{\frac{3}{3}}$ .

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

Step 1.2

Divide $3$ by $3$ .

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

Step 1.3

Simplify.

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

Step 1.4

Add $8 y^{2}$ and $y^{2}$ .

${(y^{2} - 4)}^{\frac{1}{3}} (9 y^{2} - 4) = 0$

Step 1.5

Factor.

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Step 1.5.1

Rewrite $9 y^{2} - 4$ in a factored form.

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Step 1.5.1.1

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

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

Step 1.5.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.5.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 y$ and $b = 2$ .

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

Step 1.5.2

Remove unnecessary parentheses.

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

Step 2

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

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

$3 y + 2 = 0$

$3 y - 2 = 0$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$y^{2} - 4 = 0$

Step 3.2.2

Solve for $y$ .

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Step 3.2.2.1

Add $4$ to both sides of the equation.

$y^{2} = 4$

Step 3.2.2.2

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

$y = \pm \sqrt{4}$

Step 3.2.2.3

Simplify $\pm \sqrt{4}$ .

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Step 3.2.2.3.1

Rewrite $4$ as $2^{2}$ .

$y = \pm \sqrt{2^{2}}$

Step 3.2.2.3.2

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

$y = \pm 2$

Step 3.2.2.4

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

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Step 3.2.2.4.1

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

$y = 2$

Step 3.2.2.4.2

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

$y = - 2$

Step 3.2.2.4.3

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

$y = 2, - 2$

Step 4

Set $3 y + 2$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $3 y + 2$ equal to $0$ .

$3 y + 2 = 0$

Step 4.2

Solve $3 y + 2 = 0$ for $y$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$3 y = - 2$

Step 4.2.2

Divide each term in $3 y = - 2$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 y = - 2$ by $3$ .

$\frac{3 y}{3} = \frac{- 2}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 2}{3}$

Step 4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{3}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{2}{3}$

Step 5

Set $3 y - 2$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $3 y - 2$ equal to $0$ .

$3 y - 2 = 0$

Step 5.2

Solve $3 y - 2 = 0$ for $y$ .

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Step 5.2.1

Add $2$ to both sides of the equation.

$3 y = 2$

Step 5.2.2

Divide each term in $3 y = 2$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 y = 2$ by $3$ .

$\frac{3 y}{3} = \frac{2}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2}{3}$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{3}$

Step 6

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

$y = 2, - 2, - \frac{2}{3}, \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 2, - 2, - \frac{2}{3}, \frac{2}{3}$

Decimal Form:

$y = 2, - 2, - 0.\overline{6}, 0.\overline{6}$