# If $ {\left( {\sqrt 5 + \sqrt {3i} } \right)^{33}} = {2^{49}}z $ , then modulus of the complex number is

A.1

B. $ \sqrt 2 $

C. $ 2\sqrt 2 $

D.4

Answer

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**Hint**: The complex number in the question is z so we have to find modulus of z which is $ \left| z \right| $ . For finding this we have to take modulus on both sides of the equation. To find the solution properties of complex numbers are to be used.

**:**

__Complete step-by-step answer__According to the question,

Given: $ {\left( {\sqrt 5 + \sqrt {3i} } \right)^{33}} = {2^{49}}z $

Now, we have to find modulus of z which is $ \left| z \right| $ , so

By taking modulus on both sides we get,

$ \left| {{{\left( {\sqrt 5 + \sqrt {3i} } \right)}^{33}}} \right| = \left| {{2^{49}}z} \right| $

By using the property of complex number $ \left| {{z^n}} \right| = {\left| z \right|^n} $ , we get

$ {\left| {\left( {\sqrt 5 + \sqrt {3i} } \right)} \right|^{33}} = {2^{49}}\left| z \right| $

Now using $ z = a + ib $ then, $ \left| z \right| = \sqrt {{a^2} + {b^2}} $ property of complex number, we get

$ {\left( {\sqrt {\sqrt {{{\left( 5 \right)}^2}} + \sqrt {{{\left( 3 \right)}^2}} } } \right)^{33}} = {2^{49}}\left| z \right| $

Now after solving inside bracket, we get

$ {\left( {\sqrt 8 } \right)^{33}} = {2^{49}}\left| z \right| $

$ {\left( 8 \right)^{\dfrac{{33}}{2}}} = {2^{49}}\left| z \right| $

$ {({2^3})^{\dfrac{{33}}{2}}} = {2^{49}}\left| z \right| $

$ {\left( 2 \right)^{\dfrac{{99}}{2}}} = {2^{49}}\left| z \right| $

$ \left| z \right| = {2^{\left( {\dfrac{{99}}{2} - 49} \right)}} $

$ \left| z \right| = {2^{\left( {\dfrac{{99 - 98}}{2}} \right)}} $

$ \left| z \right| = {2^{\dfrac{1}{2}}} $

$ \left| z \right| = \sqrt 2 $

The value of z modulus is $ \sqrt 2 $ . So, option (B) is the correct answer.

**So, the correct answer is “Option B”.**

**Note**: Whenever we face such types of problems we use some important points. First we find real and imaginary parts of a complex number, then apply the formula of modulus of a complex number, then after solving we can get the required answer. In this question we are required to find the modulus of z and in the question we are provided with an equation in which z is present. So, we took modulus on both sides to find the answer , the reason we took modulus is that, in the equation z was without modulus and we have to find the value of z modulus which is only possible with the way if we take modulus on both sides.

Last updated date: 28th May 2023

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