The value of 1-1-21-31-4 -1-3 n is

(1+ω)(1+ω^2)(1+ω^3)(1+ω^4)⋯(1+ω^3n) =(1+ω)(1+ω^2)(1+1)(1+ω)(1+ω^2)(1+1)⋯(1+1) =[( ω^2)( ω)]2[( ω^2)( ω)]2⋯[( ω^2)( ω)]2 [∵ 1+ω+ω^2=0] =(ω^3)·2·(ω^3)·2⋯2· (ω^ ...

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Standard XII

Mathematics

Square Root of a Complex Number

The value of ...

Question

The value of $(1 + ω) (1 + ω^{2}) (1 + ω^{3}) (1 + ω^{4}) \dots (1 + ω^{3 n})$ is

2^{3 n}

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2^{2 n}

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2^{n}

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2^{4 n}

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Solution

The correct option is C $2^{n}$
$(1 + ω) (1 + ω^{2}) (1 + ω^{3}) (1 + ω^{4}) \dots (1 + ω^{3 n})$
$= (1 + ω) (1 + ω^{2}) (1 + 1) (1 + ω) (1 + ω^{2}) (1 + 1) \dots (1 + 1) = [(- ω^{2}) (- ω)] 2 [(- ω^{2}) (- ω)] 2 \dots [(- ω^{2}) (- ω)] 2 [∵ 1 + ω + ω^{2} = 0] = (ω^{3}) \cdot 2 \cdot (ω^{3}) \cdot 2 \dots 2 \cdot (ω^{3}) \cdot 2$
$= 2 \cdot 2 \cdot 2 \dots n terms$
$= 2^{n}$

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Similar questions

Q. If

ω

is an imaginary cube root of unity, then the value of

(1 + ω) (1 + ω^{2}) (1 + ω^{3}) (1 + ω^{4}) (1 + ω^{5}) . . . . (1 + ω^{3 n})

Q. Arrange the following values in ascending order:
I)

(1 + ω) (1 + ω^{2}) (1 + ω^{4}) (1 + ω^{8})

II)

(1 - ω + ω^{2})^{7} + (1 + ω - ω^{2})^{7}

III)

(1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{8})

IV)

(1 - ω + ω^{2}) (1 - ω^{2} + ω^{4})

i = \sqrt{- 1}

ω =

nonreal cube root unity then

\frac{(1 + i)^{2 n} - (1 - i)^{2 n}}{(1 + ω^{4} - ω^{2}) (1 - ω^{4} + ω^{2})}

is equal to

Q. If

i = \sqrt{- 1}, ω

= nonreal cube root of unity then

\frac{(1 + i)^{2 n} - (1 - i)^{2 n}}{(1 + ω^{4} - ω^{2}) (1 - ω^{4} + ω^{2})}

is equal to:

(1 - ω + ω^{2}) (1 - ω^{2} + ω^{4}) (1 - ω^{4} + ω^{8}) (1 - ω^{8} + ω^{16}) =

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Square Root of a Complex Number

Standard XII Mathematics

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The value of (1+ω)(1+ω2)(1+ω3)(1+ω4)⋯(1+ω3n) is

The correct option is C 2n(1+ω)(1+ω2)(1+ω3)(1+ω4)⋯(1+ω3n) =(1+ω)(1+ω2)(1+1)(1+ω)(1+ω2)(1+1)⋯(1+1)=[(−ω2)(−ω)]2[(−ω2)(−ω)]2⋯[(−ω2)(−ω)]2[∵1+ω+ω2=0]=(ω3)⋅2⋅(ω3)⋅2⋯2⋅(ω3)⋅2 =2⋅2⋅2⋯ n terms =2n

The value of $(1 + ω) (1 + ω^{2}) (1 + ω^{3}) (1 + ω^{4}) \dots (1 + ω^{3 n})$ is