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Popular >

simplify (e^{ti}-e^{-ti})^3

Solution

simplify

(e^{t i} - e^{- t i})^{3}

Solution

- 8 i sin^{3} (t)

Solution steps

(e^{t i} - e^{- t i})^{3}

e^{t i} = cos (t) + i sin (t)

e^{- t i} = cos (t) - i sin (t)

= (cos (t) + i sin (t) - (cos (t) - i sin (t)))^{3}

Expand

cos (t) + sin (t) i - (cos (t) - i sin (t)) : 2 i sin (t)

= (2 i sin (t))^{3}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{3} i^{3} sin^{3} (t)

2^{3} = 8

= 8 i^{3} sin^{3} (t)

i^{3} = - i

= 8 (- i) sin^{3} (t)

Remove parentheses:

(- a) = - a

= - 8 i sin^{3} (t)

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Frequently Asked Questions (FAQ)

What is (e^{ti}-e^{-ti})^3 ?
The solution to (e^{ti}-e^{-ti})^3 is -8isin^3(t)