Calculus Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.2

Combine ${(16 - x^{\frac{2}{3}})}^{\frac{1}{2}}$ and $\frac{1}{x^{\frac{1}{3}}}$ .

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

Step 1.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.3.1

Apply the product rule to $- \frac{{(16 - x^{\frac{2}{3}})}^{\frac{1}{2}}}{x^{\frac{1}{3}}}$ .

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

Step 1.1.3.2

Apply the product rule to $\frac{{(16 - x^{\frac{2}{3}})}^{\frac{1}{2}}}{x^{\frac{1}{3}}}$ .

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

Step 1.1.4

Raise $- 1$ to the power of $2$ .

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

Step 1.1.5

Multiply $\frac{{({(16 - x^{\frac{2}{3}})}^{\frac{1}{2}})}^{2}}{{(x^{\frac{1}{3}})}^{2}}$ by $1$ .

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply the exponents in ${({(16 - x^{\frac{2}{3}})}^{\frac{1}{2}})}^{2}$ .

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Step 1.1.6.1.1

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

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

Step 1.1.6.1.2

Cancel the common factor of $2$ .

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Step 1.1.6.1.2.1

Cancel the common factor.

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

Step 1.1.6.1.2.2

Rewrite the expression.

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

Step 1.1.6.2

Simplify.

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

Step 1.1.6.3

Rewrite $16$ as $4^{2}$ .

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

Step 1.1.6.4

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

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

Step 1.1.6.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x^{\frac{1}{3}}$ .

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

Step 1.1.7

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 1.1.7.1

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

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

Step 1.1.7.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 1.2

Combine into one fraction.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $4$ by $4$ .

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

Step 2.2.1.2

Multiply $- 1$ by $4$ .

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

Step 2.2.1.3

Move $4$ to the left of $x^{\frac{1}{3}}$ .

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

Step 2.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.5

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

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Step 2.2.1.5.1

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

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

Step 2.2.1.5.2

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

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

Step 2.2.1.5.3

Combine the numerators over the common denominator.

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

Step 2.2.1.5.4

Add $1$ and $1$ .

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

Step 2.2.2

Add $- 4 x^{\frac{1}{3}}$ and $4 x^{\frac{1}{3}}$ .

$\frac{x^{\frac{2}{3}} + 16 + 0 - x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$

Step 2.2.3

Add $16$ and $0$ .

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

Step 2.3

Subtract $x^{\frac{2}{3}}$ from $x^{\frac{2}{3}}$ .

$\frac{0 + 16}{x^{\frac{2}{3}}}$

Step 2.4

Add $0$ and $16$ .

$\frac{16}{x^{\frac{2}{3}}}$