Calculus Examples

$f (x) = - \frac{3}{2 t^{\frac{3}{2}}}$

Step 1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{3}{2 \sqrt{t^{3}}}$

Step 2

Set the radicand in $\sqrt{t^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$t^{3} \geq 0$

Step 3

Solve for $t$ .

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Step 3.1

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

$\sqrt[3]{t^{3}} \geq \sqrt[3]{0}$

Step 3.2

Simplify the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Pull terms out from under the radical.

$t \geq \sqrt[3]{0}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$t \geq \sqrt[3]{0^{3}}$

Step 3.2.2.1.2

Pull terms out from under the radical.

$t \geq 0$

Step 4

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

$2 \sqrt{t^{3}} = 0$

Step 5

Solve for $t$ .

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Step 5.1

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

${(2 \sqrt{t^{3}})}^{2} = 0^{2}$

Step 5.2

Simplify each side of the equation.

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Step 5.2.1

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

${(2 t^{\frac{3}{2}})}^{2} = 0^{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Simplify ${(2 t^{\frac{3}{2}})}^{2}$ .

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Step 5.2.2.1.1

Apply the product rule to $2 t^{\frac{3}{2}}$ .

$2^{2} {(t^{\frac{3}{2}})}^{2} = 0^{2}$

Step 5.2.2.1.2

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

$4 {(t^{\frac{3}{2}})}^{2} = 0^{2}$

Step 5.2.2.1.3

Multiply the exponents in ${(t^{\frac{3}{2}})}^{2}$ .

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Step 5.2.2.1.3.1

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

$4 t^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.2.1

Cancel the common factor.

$4 t^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.3.2.2

Rewrite the expression.

$4 t^{3} = 0^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 t^{3} = 0$

Step 5.3

Solve for $t$ .

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Step 5.3.1

Divide each term in $4 t^{3} = 0$ by $4$ and simplify.

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Step 5.3.1.1

Divide each term in $4 t^{3} = 0$ by $4$ .

$\frac{4 t^{3}}{4} = \frac{0}{4}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $4$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{4 t^{3}}{4} = \frac{0}{4}$

Step 5.3.1.2.1.2

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

$t^{3} = \frac{0}{4}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $4$ .

$t^{3} = 0$

Step 5.3.2

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

$t = \sqrt[3]{0}$

Step 5.3.3

Simplify $\sqrt[3]{0}$ .

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Step 5.3.3.1

Rewrite $0$ as $0^{3}$ .

$t = \sqrt[3]{0^{3}}$

Step 5.3.3.2

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

$t = 0$

Step 6

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${t | t > 0}$

Step 7