Linear Algebra Examples

$y = \frac{t^{3} + t}{2 t^{3} + 1}$

Step 1

Set the denominator in $\frac{t^{3} + t}{2 t^{3} + 1}$ equal to $0$ to find where the expression is undefined.

$2 t^{3} + 1 = 0$

Step 2

Solve for $t$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$2 t^{3} = - 1$

Step 2.2

Divide each term in $2 t^{3} = - 1$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 t^{3} = - 1$ by $2$ .

$\frac{2 t^{3}}{2} = \frac{- 1}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 t^{3}}{2} = \frac{- 1}{2}$

Step 2.2.2.1.2

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

$t^{3} = \frac{- 1}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$t^{3} = - \frac{1}{2}$

Step 2.3

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

$t = \sqrt[3]{- \frac{1}{2}}$

Step 2.4

Simplify $\sqrt[3]{- \frac{1}{2}}$ .

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Step 2.4.1

Rewrite $- \frac{1}{2}$ as ${({(- 1)}^{3})}^{3} \frac{1}{2}$ .

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Step 2.4.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$t = \sqrt[3]{{(- 1)}^{3} \frac{1}{2}}$

Step 2.4.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$t = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{1}{2}}$

Step 2.4.2

Pull terms out from under the radical.

$t = {(- 1)}^{3} \sqrt[3]{\frac{1}{2}}$

Step 2.4.3

Raise $- 1$ to the power of $3$ .

$t = - \sqrt[3]{\frac{1}{2}}$

Step 2.4.4

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

$t = - \frac{\sqrt[3]{1}}{\sqrt[3]{2}}$

Step 2.4.5

Any root of $1$ is $1$ .

$t = - \frac{1}{\sqrt[3]{2}}$

Step 2.4.6

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

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

Step 2.4.7

Combine and simplify the denominator.

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Step 2.4.7.1

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

$t = - \frac{{\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 2.4.7.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$t = - \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 2.4.7.3

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

$t = - \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 2.4.7.4

Add $1$ and $2$ .

$t = - \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 2.4.7.5

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

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Step 2.4.7.5.1

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

$t = - \frac{{\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 2.4.7.5.2

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

$t = - \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 2.4.7.5.3

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

$t = - \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.4.7.5.4

Cancel the common factor of $3$ .

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Step 2.4.7.5.4.1

Cancel the common factor.

$t = - \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.4.7.5.4.2

Rewrite the expression.

$t = - \frac{{\sqrt[3]{2}}^{2}}{2^{1}}$

Step 2.4.7.5.5

Evaluate the exponent.

$t = - \frac{{\sqrt[3]{2}}^{2}}{2}$

Step 2.4.8

Simplify the numerator.

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Step 2.4.8.1

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

$t = - \frac{\sqrt[3]{2^{2}}}{2}$

Step 2.4.8.2

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

$t = - \frac{\sqrt[3]{4}}{2}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{\sqrt[3]{4}}{2}) \cup (- \frac{\sqrt[3]{4}}{2}, \infty)$

Set-Builder Notation:

${t | t \neq - \frac{\sqrt[3]{4}}{2}}$

Step 4