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Popular >

simplify (e^{it}-e^{-it})^5

Solution

simplify

(e^{i t} - e^{- i t})^{5}

Solution

32 i sin^{5} (t)

Solution steps

(e^{i t} - e^{- i t})^{5}

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

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

= (cos (t) + i sin (t) - (cos (t) - i sin (t)))^{5}

Expand

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

= (2 i sin (t))^{5}

Apply exponent rule:

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

= 2^{5} i^{5} sin^{5} (t)

2^{5} = 32

= 32 i^{5} sin^{5} (t)

i^{5} = i

= 32 i sin^{5} (t)

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Frequently Asked Questions (FAQ)

What is (e^{it}-e^{-it})^5 ?
The solution to (e^{it}-e^{-it})^5 is 32isin^5(t)