Calculus Examples

$4 x^{3} - 14$

Step 1

Set $4 x^{3} - 14$ equal to $0$ .

$4 x^{3} - 14 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $14$ to both sides of the equation.

$4 x^{3} = 14$

Step 2.2

Subtract $14$ from both sides of the equation.

$4 x^{3} - 14 = 0$

Step 2.3

Factor $2$ out of $4 x^{3} - 14$ .

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Step 2.3.1

Factor $2$ out of $4 x^{3}$ .

$2 (2 x^{3}) - 14 = 0$

Step 2.3.2

Factor $2$ out of $- 14$ .

$2 (2 x^{3}) + 2 (- 7) = 0$

Step 2.3.3

Factor $2$ out of $2 (2 x^{3}) + 2 (- 7)$ .

$2 (2 x^{3} - 7) = 0$

Step 2.4

Divide each term in $2 (2 x^{3} - 7) = 0$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 (2 x^{3} - 7) = 0$ by $2$ .

$\frac{2 (2 x^{3} - 7)}{2} = \frac{0}{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{2 (2 x^{3} - 7)}{2} = \frac{0}{2}$

Step 2.4.2.1.2

Divide $2 x^{3} - 7$ by $1$ .

$2 x^{3} - 7 = \frac{0}{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $0$ by $2$ .

$2 x^{3} - 7 = 0$

Step 2.5

Add $7$ to both sides of the equation.

$2 x^{3} = 7$

Step 2.6

Divide each term in $2 x^{3} = 7$ by $2$ and simplify.

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Step 2.6.1

Divide each term in $2 x^{3} = 7$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{7}{2}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $2$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{7}{2}$

Step 2.6.2.1.2

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

$x^{3} = \frac{7}{2}$

Step 2.7

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

$x = \sqrt[3]{\frac{7}{2}}$

Step 2.8

Simplify $\sqrt[3]{\frac{7}{2}}$ .

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Step 2.8.1

Rewrite $\sqrt[3]{\frac{7}{2}}$ as $\frac{\sqrt[3]{7}}{\sqrt[3]{2}}$ .

$x = \frac{\sqrt[3]{7}}{\sqrt[3]{2}}$

Step 2.8.2

Multiply $\frac{\sqrt[3]{7}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{\sqrt[3]{7}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 2.8.3

Combine and simplify the denominator.

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Step 2.8.3.1

Multiply $\frac{\sqrt[3]{7}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 2.8.3.2

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

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 2.8.3.3

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

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 2.8.3.4

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 2.8.3.5

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

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Step 2.8.3.5.1

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

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 2.8.3.5.2

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

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 2.8.3.5.3

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

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.8.3.5.4

Cancel the common factor of $3$ .

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Step 2.8.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.8.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 2.8.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[3]{7} {\sqrt[3]{2}}^{2}}{2}$

Step 2.8.4

Simplify the numerator.

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Step 2.8.4.1

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

$x = \frac{\sqrt[3]{7} \sqrt[3]{2^{2}}}{2}$

Step 2.8.4.2

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

$x = \frac{\sqrt[3]{7} \sqrt[3]{4}}{2}$

Step 2.8.5

Simplify the numerator.

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Step 2.8.5.1

Combine using the product rule for radicals.

$x = \frac{\sqrt[3]{7 \cdot 4}}{2}$

Step 2.8.5.2

Multiply $7$ by $4$ .

$x = \frac{\sqrt[3]{28}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[3]{28}}{2}$

Decimal Form:

$x = 1.51829448 \dots$

Step 4