Answer
385.5k+ views
Hint: Now consider the given number ${{i}^{403}}$ we know that the value of ${{i}^{4}}=1$ and ${{i}^{3}}=-i$ hence we will first simplify the power and write the number in the form of ${{i}^{3}}$ and ${{i}^{4}}$ by using the laws of indices ${{x}^{m+n}}={{x}^{m}}{{x}^{n}}$ and ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$ . Now we will substitute the values of ${{i}^{3}}$ and ${{i}^{4}}$ and hence find the solution.
Complete step by step answer:
Now let us first understand the meaning of letter i.
Now we know the number line which represents real numbers which are either rational or irrational.
But there are also numbers which are not real. These numbers are called complex numbers.
Complex numbers are numbers of the form a + ib. where a and b are real and the letter i denotes iota which is nothing but $\sqrt{-1}$ .
Now since we have $i=\sqrt{-1}$ squaring both the sides we get ${{i}^{2}}=-1$
Now multiplying I on both sides we get, ${{i}^{3}}=-i$ again multiplying i on both sides we get, ${{i}^{4}}=-i\times i=-\left( -1 \right)=1$
Hence we can say that ${{i}^{4}}=1$
Now consider the given number ${{i}^{403}}$ .
Now we know by law of indices that ${{x}^{m+n}}={{x}^{m}}{{x}^{n}}$
Hence we can write
$\Rightarrow {{i}^{403}}={{i}^{400+3}}={{i}^{400}}{{i}^{3}}$
Now again by law of indices we know that ${{x}^{mn}}={{\left( {{x}^{m}} \right)}^{n}}$ hence using this we get,
$\Rightarrow {{i}^{403}}={{\left( {{i}^{4}} \right)}^{100}}{{i}^{3}}$
Now since ${{i}^{4}}=1$ and ${{i}^{3}}=-1$ we will substitute the values in the equation,
$\Rightarrow {{i}^{403}}={{1}^{100}}\left( -i \right)=-i$
Hence the value of ${{i}^{403}}$ is – i.
Note: Note that any power to i will be either of $-1,-i,1,i$ as after ${{i}^{4}}$ same values of I will keep repeating. Hence we can find any power of i by just using the laws of indices and the known values of i. Also note that here we also get 1 and -1 as solutions which means the square of a complex number is real. Hence we can say that multiplication of two complex numbers can be real numbers.
Complete step by step answer:
Now let us first understand the meaning of letter i.
Now we know the number line which represents real numbers which are either rational or irrational.
But there are also numbers which are not real. These numbers are called complex numbers.
Complex numbers are numbers of the form a + ib. where a and b are real and the letter i denotes iota which is nothing but $\sqrt{-1}$ .
Now since we have $i=\sqrt{-1}$ squaring both the sides we get ${{i}^{2}}=-1$
Now multiplying I on both sides we get, ${{i}^{3}}=-i$ again multiplying i on both sides we get, ${{i}^{4}}=-i\times i=-\left( -1 \right)=1$
Hence we can say that ${{i}^{4}}=1$
Now consider the given number ${{i}^{403}}$ .
Now we know by law of indices that ${{x}^{m+n}}={{x}^{m}}{{x}^{n}}$
Hence we can write
$\Rightarrow {{i}^{403}}={{i}^{400+3}}={{i}^{400}}{{i}^{3}}$
Now again by law of indices we know that ${{x}^{mn}}={{\left( {{x}^{m}} \right)}^{n}}$ hence using this we get,
$\Rightarrow {{i}^{403}}={{\left( {{i}^{4}} \right)}^{100}}{{i}^{3}}$
Now since ${{i}^{4}}=1$ and ${{i}^{3}}=-1$ we will substitute the values in the equation,
$\Rightarrow {{i}^{403}}={{1}^{100}}\left( -i \right)=-i$
Hence the value of ${{i}^{403}}$ is – i.
Note: Note that any power to i will be either of $-1,-i,1,i$ as after ${{i}^{4}}$ same values of I will keep repeating. Hence we can find any power of i by just using the laws of indices and the known values of i. Also note that here we also get 1 and -1 as solutions which means the square of a complex number is real. Hence we can say that multiplication of two complex numbers can be real numbers.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)