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If ( {3}{2}+ { 3}{2})^{50} =3^{25} (x-iy) where x, y are real,

Question

If $(\frac{3}{2} + \frac{3}{2})^{50} = 3^{25} (x - i y)$ where $x, y$ are real, then the ordered pair $(x, y)$ given by

(0, 3)

(1/2, - 3 /2)

(- 3, 0)

(0, - 3)

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Sophia Discussed

(\frac{3}{2} + \frac{3}{2})^{50} = 3^{25} (x - i y)

where

x, y

are real, then the ordered pair

(x, y)

given by

15 mins ago

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15 mins ago

Text solutionVerified

We have,

(\frac{3}{2} + \frac{3}{2})^{50} = 3^{25} (x - i y)

(\frac{- i ^{2} 3 3}{2} + \frac{3}{2})^{50} = 3^{25} (x - i y)

\Rightarrow

(i 3)^{50} (\frac{1}{2} - \frac{i 3}{2})^{50} = 3^{25} (x - i y)

\Rightarrow

[(i 3)^{2}]^{25} (\frac{1}{2} - \frac{i 3}{2})^{50} = 3^{25} (x - i y)

\Rightarrow

- (3)^{25} (\frac{1}{2} - \frac{i 3}{2})^{50} = 3^{25} (x - i y)

[ Since,

i^{2} = - 1

]

\Rightarrow

- w^{50} = (x - i y)

[ Since,

w = \frac{1}{2} - \frac{i 3}{2}

]

Now,

w^{50} = (w^{3})^{16} . w^{2} = w^{2}

\Rightarrow

- w^{2} = x - i y

\Rightarrow

\frac{1}{2} + \frac{i 3}{2} = x - i y

[ Since,

w^{2} = - \frac{1}{2} - \frac{i 3}{2}

]

∴

x = \frac{1}{2}

and

y = \frac{- 3}{2}

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Question Text	If $(\frac{3}{2} + \frac{3}{2})^{50} = 3^{25} (x - i y)$ where $x, y$ are real, then the ordered pair $(x, y)$ given by
Answer Type	Text solution:1
Upvotes	150