Precalculus Examples

$f (t) = t^{- 3}$ ;, $(\frac{1}{2}, 8)$

Step 1

Write $f (t) = t^{- 3}$ as an equation.

$y = t^{- 3}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $t$ values of the two points.

$\frac{f (8) - f (\frac{1}{2})}{(8) - (\frac{1}{2})}$

Step 2.2

Substitute the equation $y = t^{- 3}$ for $f (8)$ and $f (\frac{1}{2})$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{({(8)}^{- 3}) - ({(\frac{1}{2})}^{- 3})}{(8) - (\frac{1}{2})}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

Multiply $\frac{8^{- 3} - {(\frac{1}{2})}^{- 3}}{8 - \frac{1}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{8^{- 3} - {(\frac{1}{2})}^{- 3}}{8 - \frac{1}{2}}$

Step 3.1.2

Combine.

$\frac{2 (8^{- 3} - {(\frac{1}{2})}^{- 3})}{2 (8 - \frac{1}{2})}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 8^{- 3} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 + 2 (- \frac{1}{2})}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{2 \cdot 8^{- 3} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 + 2 (\frac{- 1}{2})}$

Step 3.3.2

Cancel the common factor.

$\frac{2 \cdot 8^{- 3} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 + 2 (\frac{- 1}{2})}$

Step 3.3.3

Rewrite the expression.

$\frac{2 \cdot 8^{- 3} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

$\frac{2 \cdot \frac{1}{8^{3}} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4.2

Raise $8$ to the power of $3$ .

$\frac{2 \cdot \frac{1}{512} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4.3

Cancel the common factor of $2$ .

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Step 3.4.3.1

Factor $2$ out of $512$ .

$\frac{2 \cdot \frac{1}{2 (256)} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4.3.2

Cancel the common factor.

$\frac{2 \cdot \frac{1}{2 \cdot 256} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4.3.3

Rewrite the expression.

$\frac{\frac{1}{256} + 2 (- {(\frac{1}{2})}^{- 3})}{2 \cdot 8 - 1}$

Step 3.4.4

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{\frac{1}{256} + 2 (- 2^{3})}{2 \cdot 8 - 1}$

Step 3.4.5

Multiply $2$ by $2^{3}$ by adding the exponents.

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Step 3.4.5.1

Move $2^{3}$ .

$\frac{\frac{1}{256} + 2^{3} \cdot 2 \cdot - 1}{2 \cdot 8 - 1}$

Step 3.4.5.2

Multiply $2^{3}$ by $2$ .

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Step 3.4.5.2.1

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

$\frac{\frac{1}{256} + 2^{3} \cdot 2^{1} \cdot - 1}{2 \cdot 8 - 1}$

Step 3.4.5.2.2

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

$\frac{\frac{1}{256} + 2^{3 + 1} \cdot - 1}{2 \cdot 8 - 1}$

Step 3.4.5.3

Add $3$ and $1$ .

$\frac{\frac{1}{256} + 2^{4} \cdot - 1}{2 \cdot 8 - 1}$

Step 3.4.6

Raise $2$ to the power of $4$ .

$\frac{\frac{1}{256} + 16 \cdot - 1}{2 \cdot 8 - 1}$

Step 3.4.7

Multiply $16$ by $- 1$ .

$\frac{\frac{1}{256} - 16}{2 \cdot 8 - 1}$

Step 3.4.8

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$\frac{\frac{1}{256} - 16 \cdot \frac{256}{256}}{2 \cdot 8 - 1}$

Step 3.4.9

Combine $- 16$ and $\frac{256}{256}$ .

$\frac{\frac{1}{256} + \frac{- 16 \cdot 256}{256}}{2 \cdot 8 - 1}$

Step 3.4.10

Combine the numerators over the common denominator.

$\frac{\frac{1 - 16 \cdot 256}{256}}{2 \cdot 8 - 1}$

Step 3.4.11

Simplify the numerator.

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Step 3.4.11.1

Multiply $- 16$ by $256$ .

$\frac{\frac{1 - 4096}{256}}{2 \cdot 8 - 1}$

Step 3.4.11.2

Subtract $4096$ from $1$ .

$\frac{\frac{- 4095}{256}}{2 \cdot 8 - 1}$

Step 3.4.12

Move the negative in front of the fraction.

$\frac{- \frac{4095}{256}}{2 \cdot 8 - 1}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $2$ by $8$ .

$\frac{- \frac{4095}{256}}{16 - 1}$

Step 3.5.2

Subtract $1$ from $16$ .

$\frac{- \frac{4095}{256}}{15}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{4095}{256} \cdot \frac{1}{15}$

Step 3.7

Cancel the common factor of $15$ .

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Step 3.7.1

Move the leading negative in $- \frac{4095}{256}$ into the numerator.

$\frac{- 4095}{256} \cdot \frac{1}{15}$

Step 3.7.2

Factor $15$ out of $- 4095$ .

$\frac{15 (- 273)}{256} \cdot \frac{1}{15}$

Step 3.7.3

Cancel the common factor.

$\frac{15 \cdot - 273}{256} \cdot \frac{1}{15}$

Step 3.7.4

Rewrite the expression.

$\frac{- 273}{256}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{273}{256}$