Calculus Examples

${(x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})}^{2}$

Step 1

Rewrite ${(x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})}^{2}$ as $(x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}}) (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})$ .

$(x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}}) (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})$

Step 2

Expand $(x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}}) (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x^{\frac{1}{3}} (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})$

Step 2.2

Apply the distributive property.

$x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} (- \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} (x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}})$

Step 2.3

Apply the distributive property.

$x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} (- \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 3.1.1.1

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

$x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} (- \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.1.2

Combine the numerators over the common denominator.

$x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} (- \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.1.3

Add $1$ and $1$ .

$x^{\frac{2}{3}} + x^{\frac{1}{3}} (- \frac{1}{4 x^{\frac{1}{3}}}) - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$x^{\frac{2}{3}} - x^{\frac{1}{3}} \frac{1}{4 x^{\frac{1}{3}}} - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.3

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 3.1.3.1

Factor $x^{\frac{1}{3}}$ out of $- x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot - 1 \frac{1}{4 x^{\frac{1}{3}}} - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.3.2

Factor $x^{\frac{1}{3}}$ out of $4 x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot - 1 \frac{1}{x^{\frac{1}{3}} \cdot 4} - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.3.3

Cancel the common factor.

$x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot - 1 \frac{1}{x^{\frac{1}{3}} \cdot 4} - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.3.4

Rewrite the expression.

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.4

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{4 x^{\frac{1}{3}}}$ into the numerator.

$x^{\frac{2}{3}} - \frac{1}{4} + \frac{- 1}{4 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.4.2

Factor $x^{\frac{1}{3}}$ out of $4 x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} - \frac{1}{4} + \frac{- 1}{x^{\frac{1}{3}} \cdot 4} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.4.3

Cancel the common factor.

$x^{\frac{2}{3}} - \frac{1}{4} + \frac{- 1}{x^{\frac{1}{3}} \cdot 4} x^{\frac{1}{3}} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.4.4

Rewrite the expression.

$x^{\frac{2}{3}} - \frac{1}{4} + \frac{- 1}{4} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.5

Move the negative in front of the fraction.

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} - \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$

Step 3.1.6

Multiply $- \frac{1}{4 x^{\frac{1}{3}}} (- \frac{1}{4 x^{\frac{1}{3}}})$ .

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Step 3.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + 1 \frac{1}{4 x^{\frac{1}{3}}} \frac{1}{4 x^{\frac{1}{3}}}$

Step 3.1.6.2

Multiply $\frac{1}{4 x^{\frac{1}{3}}}$ by $1$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 x^{\frac{1}{3}}} \cdot \frac{1}{4 x^{\frac{1}{3}}}$

Step 3.1.6.3

Multiply $\frac{1}{4 x^{\frac{1}{3}}}$ by $\frac{1}{4 x^{\frac{1}{3}}}$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 x^{\frac{1}{3}} (4 x^{\frac{1}{3}})}$

Step 3.1.6.4

Multiply $4$ by $4$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{\frac{1}{3}} x^{\frac{1}{3}}}$

Step 3.1.6.5

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 3.1.6.5.1

Move $x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 (x^{\frac{1}{3}} x^{\frac{1}{3}})}$

Step 3.1.6.5.2

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

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{\frac{1}{3} + \frac{1}{3}}}$

Step 3.1.6.5.3

Combine the numerators over the common denominator.

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{\frac{1 + 1}{3}}}$

Step 3.1.6.5.4

Add $1$ and $1$ .

$x^{\frac{2}{3}} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{\frac{2}{3}}}$

Step 3.2

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$x^{\frac{2}{3}} - 2 (\frac{1}{4}) + \frac{1}{16 x^{\frac{2}{3}}}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$x^{\frac{2}{3}} + 2 (- 1) \frac{1}{4} + \frac{1}{16 x^{\frac{2}{3}}}$

Step 4.1.2

Factor $2$ out of $4$ .

$x^{\frac{2}{3}} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{\frac{2}{3}}}$

Step 4.1.3

Cancel the common factor.

$x^{\frac{2}{3}} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{\frac{2}{3}}}$

Step 4.1.4

Rewrite the expression.

$x^{\frac{2}{3}} - 1 (\frac{1}{2}) + \frac{1}{16 x^{\frac{2}{3}}}$

Step 4.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16 x^{\frac{2}{3}}}$