Calculus Examples

$2 x^{3} - 3 p^{4} = 6$

Step 1

Add $3 p^{4}$ to both sides of the equation.

$2 x^{3} = 6 + 3 p^{4}$

Step 2

Divide each term in $2 x^{3} = 6 + 3 p^{4}$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x^{3} = 6 + 3 p^{4}$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{6}{2} + \frac{3 p^{4}}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{6}{2} + \frac{3 p^{4}}{2}$

Step 2.2.1.2

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

$x^{3} = \frac{6}{2} + \frac{3 p^{4}}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $6$ by $2$ .

$x^{3} = 3 + \frac{3 p^{4}}{2}$

Step 3

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

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

Step 4

Simplify $\sqrt[3]{3 + \frac{3 p^{4}}{2}}$ .

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Step 4.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2

Combine $3$ and $\frac{2}{2}$ .

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Factor $3$ out of $3 \cdot 2 + 3 p^{4}$ .

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Step 4.4.1

Factor $3$ out of $3 \cdot 2$ .

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

Step 4.4.2

Factor $3$ out of $3 p^{4}$ .

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

Step 4.4.3

Factor $3$ out of $3 (2) + 3 (p^{4})$ .

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

Step 4.5

Rewrite $\sqrt[3]{\frac{3 (2 + p^{4})}{2}}$ as $\frac{\sqrt[3]{3 (2 + p^{4})}}{\sqrt[3]{2}}$ .

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

Step 4.6

Multiply $\frac{\sqrt[3]{3 (2 + p^{4})}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{\sqrt[3]{3 (2 + p^{4})}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{\sqrt[3]{3 (2 + p^{4})}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Add $1$ and $2$ .

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

Step 4.7.5

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

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Step 4.7.5.1

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

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

Step 4.7.5.2

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

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

Step 4.7.5.3

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

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

Step 4.7.5.4

Cancel the common factor of $3$ .

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Step 4.7.5.4.1

Cancel the common factor.

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

Step 4.7.5.4.2

Rewrite the expression.

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

Step 4.7.5.5

Evaluate the exponent.

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

Step 4.8

Simplify the numerator.

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Step 4.8.1

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

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

Step 4.8.2

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

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

Step 4.9

Simplify the numerator.

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Step 4.9.1

Combine using the product rule for radicals.

$x = \frac{\sqrt[3]{3 (2 + p^{4}) \cdot 4}}{2}$

Step 4.9.2

Multiply $4$ by $3$ .

$x = \frac{\sqrt[3]{12 (2 + p^{4})}}{2}$