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Popular >

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

Solution

simplify

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

Solution

- 8 i sin^{3} (x)

Solution steps

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

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

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

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

Expand

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

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

Apply exponent rule:

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

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

2^{3} = 8

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

i^{3} = - i

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

Remove parentheses:

(- a) = - a

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

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Frequently Asked Questions (FAQ)

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