Calculus Examples

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

Step 1

Rewrite the equation as $\frac{4}{t^{3}} = w$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$t^{3}$

Step 3

Multiply each term in $\frac{4}{t^{3}} = w$ by $t^{3}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{4}{t^{3}} = w$ by $t^{3}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $t^{3}$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

$4 = w t^{3}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $w t^{3} = 4$ .

$w t^{3} = 4$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $w$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

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

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

Step 4.3

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

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

Step 4.4

Simplify $\sqrt[3]{\frac{4}{w}}$ .

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Step 4.4.1

Rewrite $\sqrt[3]{\frac{4}{w}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{w}}$ .

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

Step 4.4.2

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{w}}$ by $\frac{{\sqrt[3]{w}}^{2}}{{\sqrt[3]{w}}^{2}}$ .

$t = \frac{\sqrt[3]{4}}{\sqrt[3]{w}} \cdot \frac{{\sqrt[3]{w}}^{2}}{{\sqrt[3]{w}}^{2}}$

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{w}}$ by $\frac{{\sqrt[3]{w}}^{2}}{{\sqrt[3]{w}}^{2}}$ .

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

Step 4.4.3.2

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

$t = \frac{\sqrt[3]{4} {\sqrt[3]{w}}^{2}}{{\sqrt[3]{w}}^{1} {\sqrt[3]{w}}^{2}}$

Step 4.4.3.3

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

$t = \frac{\sqrt[3]{4} {\sqrt[3]{w}}^{2}}{{\sqrt[3]{w}}^{1 + 2}}$

Step 4.4.3.4

Add $1$ and $2$ .

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

Step 4.4.3.5

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

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Step 4.4.3.5.1

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

$t = \frac{\sqrt[3]{4} {\sqrt[3]{w}}^{2}}{{(w^{\frac{1}{3}})}^{3}}$

Step 4.4.3.5.2

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

$t = \frac{\sqrt[3]{4} {\sqrt[3]{w}}^{2}}{w^{\frac{1}{3} \cdot 3}}$

Step 4.4.3.5.3

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

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

Step 4.4.3.5.4

Cancel the common factor of $3$ .

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Step 4.4.3.5.4.1

Cancel the common factor.

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

Step 4.4.3.5.4.2

Rewrite the expression.

$t = \frac{\sqrt[3]{4} {\sqrt[3]{w}}^{2}}{w^{1}}$

Step 4.4.3.5.5

Simplify.

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

Step 4.4.4

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

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

Step 4.4.5

Combine using the product rule for radicals.

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

Step 5

To rewrite as a function of $w$ , write the equation so that $t$ is by itself on one side of the equal sign and an expression involving only $w$ is on the other side.

$f (w) = \frac{\sqrt[3]{4 w^{2}}}{w}$