Simplify square root of 2+e^(2t)+e^(-2t)

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

Step 1

Factor using the perfect square rule.

Rearrange terms.

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

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

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

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

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

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

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

Rewrite the polynomial.

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

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

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

Step 2

Pull terms out from under the radical, assuming positive real numbers.

$e^{t} + e^{- t}$