Finite Math Examples

$\sqrt[3]{m} = m$

Step 1

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

${\sqrt[3]{m}}^{3} = m^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${(m^{\frac{1}{3}})}^{3} = m^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(m^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${(m^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

$m^{\frac{1}{3} \cdot 3} = m^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$m^{\frac{1}{3} \cdot 3} = m^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

$m^{1} = m^{3}$

Step 2.2.1.2

Simplify.

$m = m^{3}$

Step 3

Solve for $m$ .

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Step 3.1

Subtract $m^{3}$ from both sides of the equation.

$m - m^{3} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $m$ out of $m - m^{3}$ .

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Step 3.2.1.1

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

$m - m^{3} = 0$

Step 3.2.1.2

Factor $m$ out of $m^{1}$ .

$m \cdot 1 - m^{3} = 0$

Step 3.2.1.3

Factor $m$ out of $- m^{3}$ .

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

Step 3.2.1.4

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

$m (1 - m^{2}) = 0$

Step 3.2.2

Rewrite $1$ as $1^{2}$ .

$m (1^{2} - m^{2}) = 0$

Step 3.2.3

Factor.

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Step 3.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = m$ .

$m ((1 + m) (1 - m)) = 0$

Step 3.2.3.2

Remove unnecessary parentheses.

$m (1 + m) (1 - m) = 0$

Step 3.3

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

$m = 0$

$1 + m = 0$

$1 - m = 0$

Step 3.4

Set $m$ equal to $0$ .

$m = 0$

Step 3.5

Set $1 + m$ equal to $0$ and solve for $m$ .

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Step 3.5.1

Set $1 + m$ equal to $0$ .

$1 + m = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$m = - 1$

Step 3.6

Set $1 - m$ equal to $0$ and solve for $m$ .

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Step 3.6.1

Set $1 - m$ equal to $0$ .

$1 - m = 0$

Step 3.6.2

Solve $1 - m = 0$ for $m$ .

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Step 3.6.2.1

Subtract $1$ from both sides of the equation.

$- m = - 1$

Step 3.6.2.2

Divide each term in $- m = - 1$ by $- 1$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- m = - 1$ by $- 1$ .

$\frac{- m}{- 1} = \frac{- 1}{- 1}$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{m}{1} = \frac{- 1}{- 1}$

Step 3.6.2.2.2.2

Divide $m$ by $1$ .

$m = \frac{- 1}{- 1}$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Divide $- 1$ by $- 1$ .

$m = 1$

Step 3.7

The final solution is all the values that make $m (1 + m) (1 - m) = 0$ true.

$m = 0, - 1, 1$