Question

The modulus of i7 + i10 =​

Answer 1

Concept:

Modulus of a complex number is given as,

Consider a complex number,

z=x+iy

then modulus of this complex number would be

|z|=$\sqrt{x^{2} +y^{2} }$

Also,

$i^{4n} =1\\i^{4n+1} =i\\i^{4n+2} =-1\\i^{4n+3} =-i$

Given:

Complex number:

$i^{7} +i^{10}$

To find:

The modulus of the complex number: $i^{7} +i^{10}$

Solution:

Now we know that

$i^{7}=x^{4(1)+3}$             [it is in the form of  $i^{4n+3} =-i$]

⇒$i^{7} =-i$

Also,

$i^{10}=x^{4(2)+2}$            [it is in the form of $i^{4n+2} =-1$]

⇒$i^{10}=-1$

The simplified complex number z is

$z=-i-1$

Now, modulus od z will be

$|z|=\sqrt{(-1)^{2}+(-1)^{2} } \\|z|=\sqrt{1+1} \\|z|=\sqrt{2}$

Therefore modulus of $i^{7} +i^{10}$ is $\sqrt{2}$