Precalculus Examples

$3280 {(1.0117)}^{12 t}$

Step 1

Write $3280 {(1.0117)}^{12 t}$ as a function.

$f (t) = 3280 {(1.0117)}^{12 t}$

Step 2

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = t + h$ .

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Step 3.1.1

Replace the variable $t$ with $t + h$ in the expression.

$f (t + h) = 3280 {(1.0117)}^{12 (t + h)}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (t + h) = 3280 \cdot {1.0117}^{12 t + 12 h}$

Step 3.1.2.2

The final answer is $3280 \cdot {1.0117}^{12 t + 12 h}$ .

$3280 \cdot {1.0117}^{12 t + 12 h}$

Step 3.2

Find the components of the definition.

$f (t + h) = 3280 \cdot {1.0117}^{12 t + 12 h}$

$f (t) = 3280 \cdot {1.0117}^{12 t}$

$f (t + h) = 3280 \cdot {1.0117}^{12 t + 12 h}$

$f (t) = 3280 \cdot {1.0117}^{12 t}$

Step 4

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{3280 \cdot {1.0117}^{12 t + 12 h} - (3280 \cdot {1.0117}^{12 t})}{h}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3280$ .

$\frac{3280 \cdot {1.0117}^{12 t + 12 h} - 3280 \cdot {1.0117}^{12 t}}{h}$

Step 5.2

Rewrite $3280 \cdot {1.0117}^{12 t + 12 h} - 3280 \cdot {1.0117}^{12 t}$ in a factored form.

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Step 5.2.1

Factor $3280$ out of $3280 \cdot {1.0117}^{12 t + 12 h} - 3280 \cdot {1.0117}^{12 t}$ .

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Step 5.2.1.1

Factor $3280$ out of $3280 \cdot {1.0117}^{12 t + 12 h}$ .

$\frac{3280 \cdot {1.0117}^{12 t + 12 h} - 3280 \cdot {1.0117}^{12 t}}{h}$

Step 5.2.1.2

Factor $3280$ out of $- 3280 \cdot {1.0117}^{12 t}$ .

$\frac{3280 \cdot {1.0117}^{12 t + 12 h} + 3280 (- {1.0117}^{12 t})}{h}$

Step 5.2.1.3

Factor $3280$ out of $3280 \cdot {1.0117}^{12 t + 12 h} + 3280 (- {1.0117}^{12 t})$ .

$\frac{3280 ({1.0117}^{12 t + 12 h} - {1.0117}^{12 t})}{h}$

Step 5.2.2

Rewrite ${1.0117}^{12 t + 12 h}$ as ${({1.0117}^{4 t + 4 h})}^{3}$ .

$\frac{3280 ({({1.0117}^{4 t + 4 h})}^{3} - {1.0117}^{12 t})}{h}$

Step 5.2.3

Rewrite ${1.0117}^{12 t}$ as ${({1.0117}^{4 t})}^{3}$ .

$\frac{3280 ({({1.0117}^{4 t + 4 h})}^{3} - {({1.0117}^{4 t})}^{3})}{h}$

Step 5.2.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {1.0117}^{4 t + 4 h}$ and $b = {1.0117}^{4 t}$ .

$\frac{3280 (({1.0117}^{4 t + 4 h} - {1.0117}^{4 t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5

Simplify.

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Step 5.2.5.1

Rewrite ${1.0117}^{4 t + 4 h}$ as ${({1.0117}^{2 t + 2 h})}^{2}$ .

$\frac{3280 (({({1.0117}^{2 t + 2 h})}^{2} - {1.0117}^{4 t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.2

Rewrite ${1.0117}^{4 t}$ as ${({1.0117}^{2 t})}^{2}$ .

$\frac{3280 (({({1.0117}^{2 t + 2 h})}^{2} - {({1.0117}^{2 t})}^{2}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {1.0117}^{2 t + 2 h}$ and $b = {1.0117}^{2 t}$ .

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{2 t + 2 h} - {1.0117}^{2 t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.4

Simplify.

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Step 5.2.5.4.1

Rewrite ${1.0117}^{2 t + 2 h}$ as ${({1.0117}^{t + h})}^{2}$ .

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({({1.0117}^{t + h})}^{2} - {1.0117}^{2 t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.4.2

Rewrite ${1.0117}^{2 t}$ as ${({1.0117}^{t})}^{2}$ .

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({({1.0117}^{t + h})}^{2} - {({1.0117}^{t})}^{2}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {1.0117}^{t + h}$ and $b = {1.0117}^{t}$ .

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h} \cdot {1.0117}^{4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.5

Multiply ${1.0117}^{4 t + 4 h}$ by ${1.0117}^{4 t}$ by adding the exponents.

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Step 5.2.5.5.1

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

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{4 t + 4 h + 4 t} + {({1.0117}^{4 t})}^{2}))}{h}$

Step 5.2.5.5.2

Add $4 t$ and $4 t$ .

$\frac{3280 (({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2}))}{h}$

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({({1.0117}^{4 t + 4 h})}^{2} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2})}{h}$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply the exponents in ${({1.0117}^{4 t + 4 h})}^{2}$ .

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Step 5.3.1.1

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

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{(4 t + 4 h) \cdot 2} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2})}{h}$

Step 5.3.1.2

Apply the distributive property.

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{4 t \cdot 2 + 4 h \cdot 2} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2})}{h}$

Step 5.3.1.3

Multiply $2$ by $4$ .

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{8 t + 4 h \cdot 2} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2})}{h}$

Step 5.3.1.4

Multiply $2$ by $4$ .

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{8 t + 8 h} + {1.0117}^{8 t + 4 h} + {({1.0117}^{4 t})}^{2})}{h}$

Step 5.3.2

Multiply the exponents in ${({1.0117}^{4 t})}^{2}$ .

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Step 5.3.2.1

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

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{8 t + 8 h} + {1.0117}^{8 t + 4 h} + {1.0117}^{4 t \cdot 2})}{h}$

Step 5.3.2.2

Multiply $2$ by $4$ .

$\frac{3280 ({1.0117}^{2 t + 2 h} + {1.0117}^{2 t}) ({1.0117}^{t + h} + {1.0117}^{t}) ({1.0117}^{t + h} - {1.0117}^{t}) ({1.0117}^{8 t + 8 h} + {1.0117}^{8 t + 4 h} + {1.0117}^{8 t})}{h}$

Step 6