Finite Math Examples

$\frac{e^{4 t} + 2 e^{3 t} + e^{2 t}}{e^{2 t}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{{(e^{2 t})}^{2} + 2 e^{3 t} + e^{2 t}}{e^{2 t}}$

Step 1.1.2

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

$\frac{{(e^{2 t})}^{2} + 2 e^{3 t} + {(e^{t})}^{2}}{e^{2 t}}$

Step 1.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 e^{3 t} = 2 \cdot e^{2 t} \cdot e^{t}$

Step 1.1.4

Rewrite the polynomial.

$\frac{{(e^{2 t})}^{2} + 2 \cdot e^{2 t} \cdot e^{t} + {(e^{t})}^{2}}{e^{2 t}}$

Step 1.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = e^{2 t}$ and $b = e^{t}$ .

$\frac{{(e^{2 t} + e^{t})}^{2}}{e^{2 t}}$

Step 1.2

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

$\frac{{({(e^{t})}^{2} + e^{t})}^{2}}{e^{2 t}}$

Step 1.3

Let $u = e^{t}$ . Substitute $u$ for all occurrences of $e^{t}$ .

$\frac{{(u^{2} + u)}^{2}}{e^{2 t}}$

Step 1.4

Factor $u$ out of $u^{2} + u$ .

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Step 1.4.1

Factor $u$ out of $u^{2}$ .

$\frac{{(u \cdot u + u)}^{2}}{e^{2 t}}$

Step 1.4.2

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

$\frac{{(u \cdot u + u^{1})}^{2}}{e^{2 t}}$

Step 1.4.3

Factor $u$ out of $u^{1}$ .

$\frac{{(u \cdot u + u \cdot 1)}^{2}}{e^{2 t}}$

Step 1.4.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

$\frac{{(u (u + 1))}^{2}}{e^{2 t}}$

Step 1.5

Replace all occurrences of $u$ with $e^{t}$ .

$\frac{{(e^{t} (e^{t} + 1))}^{2}}{e^{2 t}}$

Step 1.6

Apply the product rule to $e^{t} (e^{t} + 1)$ .

$\frac{{(e^{t})}^{2} {(e^{t} + 1)}^{2}}{e^{2 t}}$

Step 1.7

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

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Step 1.7.1

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

$\frac{e^{t \cdot 2} {(e^{t} + 1)}^{2}}{e^{2 t}}$

Step 1.7.2

Move $2$ to the left of $t$ .

$\frac{e^{2 t} {(e^{t} + 1)}^{2}}{e^{2 t}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $e^{2 t}$ .

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Step 2.1.1

Cancel the common factor.

$\frac{e^{2 t} {(e^{t} + 1)}^{2}}{e^{2 t}}$

Step 2.1.2

Divide ${(e^{t} + 1)}^{2}$ by $1$ .

${(e^{t} + 1)}^{2}$

Step 2.2

Rewrite ${(e^{t} + 1)}^{2}$ as $(e^{t} + 1) (e^{t} + 1)$ .

$(e^{t} + 1) (e^{t} + 1)$

Step 3

Expand $(e^{t} + 1) (e^{t} + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$e^{t} (e^{t} + 1) + 1 (e^{t} + 1)$

Step 3.2

Apply the distributive property.

$e^{t} e^{t} + e^{t} \cdot 1 + 1 (e^{t} + 1)$

Step 3.3

Apply the distributive property.

$e^{t} e^{t} + e^{t} \cdot 1 + 1 e^{t} + 1 \cdot 1$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $e^{t}$ by $e^{t}$ by adding the exponents.

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Step 4.1.1.1

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

$e^{t + t} + e^{t} \cdot 1 + 1 e^{t} + 1 \cdot 1$

Step 4.1.1.2

Add $t$ and $t$ .

$e^{2 t} + e^{t} \cdot 1 + 1 e^{t} + 1 \cdot 1$

Step 4.1.2

Multiply $e^{t}$ by $1$ .

$e^{2 t} + e^{t} + 1 e^{t} + 1 \cdot 1$

Step 4.1.3

Multiply $e^{t}$ by $1$ .

$e^{2 t} + e^{t} + e^{t} + 1 \cdot 1$

Step 4.1.4

Multiply $1$ by $1$ .

$e^{2 t} + e^{t} + e^{t} + 1$

Step 4.2

Add $e^{t}$ and $e^{t}$ .

$e^{2 t} + 2 e^{t} + 1$