Finite Math Examples

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

Step 1

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

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

Step 2

Expand $(e^{2 t} + 2 + e^{- 2 t}) (e^{2 t} + 2 + e^{- 2 t})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $e^{2 t}$ by $e^{2 t}$ by adding the exponents.

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Step 3.1.1.1

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

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

Step 3.1.1.2

Add $2 t$ and $2 t$ .

$e^{4 t} + e^{2 t} \cdot 2 + e^{2 t} e^{- 2 t} + 2 e^{2 t} + 2 \cdot 2 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.2

Move $2$ to the left of $e^{2 t}$ .

$e^{4 t} + 2 \cdot e^{2 t} + e^{2 t} e^{- 2 t} + 2 e^{2 t} + 2 \cdot 2 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.3

Multiply $e^{2 t}$ by $e^{- 2 t}$ by adding the exponents.

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Step 3.1.3.1

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

$e^{4 t} + 2 e^{2 t} + e^{2 t - 2 t} + 2 e^{2 t} + 2 \cdot 2 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.3.2

Subtract $2 t$ from $2 t$ .

$e^{4 t} + 2 e^{2 t} + e^{0} + 2 e^{2 t} + 2 \cdot 2 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.4

Simplify $e^{0}$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 2 \cdot 2 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.5

Multiply $2$ by $2$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + e^{- 2 t} e^{2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.6

Multiply $e^{- 2 t}$ by $e^{2 t}$ by adding the exponents.

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Step 3.1.6.1

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

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + e^{- 2 t + 2 t} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.6.2

Add $- 2 t$ and $2 t$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + e^{0} + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.7

Simplify $e^{0}$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + 1 + e^{- 2 t} \cdot 2 + e^{- 2 t} e^{- 2 t}$

Step 3.1.8

Move $2$ to the left of $e^{- 2 t}$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + 1 + 2 \cdot e^{- 2 t} + e^{- 2 t} e^{- 2 t}$

Step 3.1.9

Multiply $e^{- 2 t}$ by $e^{- 2 t}$ by adding the exponents.

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Step 3.1.9.1

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

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + 1 + 2 e^{- 2 t} + e^{- 2 t - 2 t}$

Step 3.1.9.2

Subtract $2 t$ from $- 2 t$ .

$e^{4 t} + 2 e^{2 t} + 1 + 2 e^{2 t} + 4 + 2 e^{- 2 t} + 1 + 2 e^{- 2 t} + e^{- 4 t}$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $2 e^{2 t}$ and $2 e^{2 t}$ .

$e^{4 t} + 4 e^{2 t} + 1 + 4 + 2 e^{- 2 t} + 1 + 2 e^{- 2 t} + e^{- 4 t}$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $1$ and $4$ .

$e^{4 t} + 4 e^{2 t} + 5 + 2 e^{- 2 t} + 1 + 2 e^{- 2 t} + e^{- 4 t}$

Step 3.2.2.2

Add $5$ and $1$ .

$e^{4 t} + 4 e^{2 t} + 2 e^{- 2 t} + 6 + 2 e^{- 2 t} + e^{- 4 t}$

Step 3.2.3

Add $2 e^{- 2 t}$ and $2 e^{- 2 t}$ .

$e^{4 t} + 4 e^{2 t} + 4 e^{- 2 t} + 6 + e^{- 4 t}$