Calculus Examples

$\frac{15}{4} = x^{3}$

Step 1

Rewrite the equation as $x^{3} = \frac{15}{4}$ .

$x^{3} = \frac{15}{4}$

Step 2

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

$x = \sqrt[3]{\frac{15}{4}}$

Step 3

Simplify $\sqrt[3]{\frac{15}{4}}$ .

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Step 3.1

Rewrite $\sqrt[3]{\frac{15}{4}}$ as $\frac{\sqrt[3]{15}}{\sqrt[3]{4}}$ .

$x = \frac{\sqrt[3]{15}}{\sqrt[3]{4}}$

Step 3.2

Multiply $\frac{\sqrt[3]{15}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{15}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt[3]{15}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.3.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.3.3

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

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.3.5

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

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Step 3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.3.5.2

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

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.3.5.3

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

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.3.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{15} {\sqrt[3]{4}}^{2}}{4}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$x = \frac{\sqrt[3]{15} \sqrt[3]{4^{2}}}{4}$

Step 3.4.2

Raise $4$ to the power of $2$ .

$x = \frac{\sqrt[3]{15} \sqrt[3]{16}}{4}$

Step 3.4.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.3.1

Factor $8$ out of $16$ .

$x = \frac{\sqrt[3]{15} \sqrt[3]{8 (2)}}{4}$

Step 3.4.3.2

Rewrite $8$ as $2^{3}$ .

$x = \frac{\sqrt[3]{15} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.4

Pull terms out from under the radical.

$x = \frac{\sqrt[3]{15} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.5

Combine exponents.

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Step 3.4.5.1

Combine using the product rule for radicals.

$x = \frac{2 \sqrt[3]{15 \cdot 2}}{4}$

Step 3.4.5.2

Multiply $15$ by $2$ .

$x = \frac{2 \sqrt[3]{30}}{4}$

Step 3.5

Cancel the common factor of $2$ and $4$ .

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Step 3.5.1

Factor $2$ out of $2 \sqrt[3]{30}$ .

$x = \frac{2 (\sqrt[3]{30})}{4}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \sqrt[3]{30}}{2 \cdot 2}$

Step 3.5.2.2

Cancel the common factor.

$x = \frac{2 \sqrt[3]{30}}{2 \cdot 2}$

Step 3.5.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{30}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[3]{30}}{2}$

Decimal Form:

$x = 1.55361625 \dots$