Question

can anyone gave me answer of this???​

Answer 1

Answer:

I¹⁰=-1

i²⁰=1

i³⁰=-1

1+(-1)+1+(-1)=0

0 is a real number,

So, we can say that the value of 1+i¹⁰+i²⁰+i³⁰ is a real number.

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Answer 2

Answer:

To prove this is a real number firstly we should simplify it by expanding powers of ¡ .

At last it will give a real number so we will conclude that the given number is a real number .

powers of ¡ can be solved as:-》

$1 + { \iota}^{10} + { \iota}^{20} + { \iota}^{30} \\ = 1 + { ( { \iota}^{2} })^{5} + {{ (\iota}^{4})}^{5} + { { (\iota}^{2} )}^{15}$

As we know that

${ \iota}^{2} = - 1 \: and \\ \: { \iota}^{4} = 1$

So ,

$1 + { ( { \iota}^{2} })^{5} + {{ (\iota}^{4})}^{5} + { { (\iota}^{2} )}^{15} = \\ 1 + {( - 1)}^{5} + {(1)}^{5} + {( - 1)}^{15} \\ = 1 - 1 + 1 - 1 \\ = 0 \: \: \: \: \: \: \: \: \: Which \: is \: a \: real \: number</p><p> \:$

So it is true that the given expression is a real number.