Calculus Examples

${(512 + 0)}^{- \frac{1}{3}} + (- \frac{{(512 + 0)}^{- \frac{4}{3}}}{3}) (x - 0)$

Step 1

Combine the opposite terms in ${(512 + 0)}^{- \frac{1}{3}} + (- \frac{{(512 + 0)}^{- \frac{4}{3}}}{3}) (x - 0)$ .

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Step 1.1

Add $512$ and $0$ .

$512^{- \frac{1}{3}} + (- \frac{{(512 + 0)}^{- \frac{4}{3}}}{3}) (x - 0)$

Step 1.2

Add $512$ and $0$ .

$512^{- \frac{1}{3}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2

Simplify each term.

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Step 2.1

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

$\frac{1}{512^{\frac{1}{3}}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $512$ as $8^{3}$ .

$\frac{1}{{(8^{3})}^{\frac{1}{3}}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.2.2

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

$\frac{1}{8^{3 (\frac{1}{3})}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

$\frac{1}{8^{3 (\frac{1}{3})}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.2.3.2

Rewrite the expression.

$\frac{1}{8^{1}} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.2.4

Evaluate the exponent.

$\frac{1}{8} - \frac{512^{- \frac{4}{3}}}{3} (x - 0)$

Step 2.3

Simplify the numerator.

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Step 2.3.1

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

$\frac{1}{8} - \frac{\frac{1}{512^{\frac{4}{3}}}}{3} (x - 0)$

Step 2.3.2

Simplify the denominator.

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Step 2.3.2.1

Rewrite $512$ as $8^{3}$ .

$\frac{1}{8} - \frac{\frac{1}{{(8^{3})}^{\frac{4}{3}}}}{3} (x - 0)$

Step 2.3.2.2

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

$\frac{1}{8} - \frac{\frac{1}{8^{3 (\frac{4}{3})}}}{3} (x - 0)$

Step 2.3.2.3

Cancel the common factor of $3$ .

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Step 2.3.2.3.1

Cancel the common factor.

$\frac{1}{8} - \frac{\frac{1}{8^{3 (\frac{4}{3})}}}{3} (x - 0)$

Step 2.3.2.3.2

Rewrite the expression.

$\frac{1}{8} - \frac{\frac{1}{8^{4}}}{3} (x - 0)$

Step 2.3.2.4

Raise $8$ to the power of $4$ .

$\frac{1}{8} - \frac{\frac{1}{4096}}{3} (x - 0)$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{8} - (\frac{1}{4096} \cdot \frac{1}{3}) (x - 0)$

Step 2.5

Multiply $\frac{1}{4096} \cdot \frac{1}{3}$ .

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Step 2.5.1

Multiply $\frac{1}{4096}$ by $\frac{1}{3}$ .

$\frac{1}{8} - \frac{1}{4096 \cdot 3} (x - 0)$

Step 2.5.2

Multiply $4096$ by $3$ .

$\frac{1}{8} - \frac{1}{12288} (x - 0)$

Step 2.6

Subtract $0$ from $x$ .

$\frac{1}{8} - \frac{1}{12288} x$

Step 2.7

Combine $x$ and $\frac{1}{12288}$ .

$\frac{1}{8} - \frac{x}{12288}$