Question

If we have an expression as \[Z = \dfrac{{{{\left( {\sqrt 3 + i} \right)}^3}{{\left( {3i + 4} \right)}^2}}}{{{{\left( {8 + 6i} \right)}^2}}}\] then \[\left| Z \right|\] is equal to

Answer 1

Hint: A complex number is a number that can be expressed in the form \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] is a symbol called the imaginary unit, and satisfying the equation \[{i^2} = - 1\] . Because no "real" number satisfies this equation \[i\] was called an imaginary number. For a complex number \[x + iy\] , \[x\] is called the real part and \[y\] is called the imaginary part.

Complete step-by-step solution:
Complex numbers have a similar definition of equality to real numbers; two complex numbers are equal if and only if both their real and imaginary parts are equal. Addition, subtraction and multiplication of the complex numbers can be naturally defined by using the rule $i^2=−1$ combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. Modulus of the complex number is the distance of the point on the argand plane representing the complex number z from the origin.
Let \[z = x + iy\] be a complex number.
Then modulus of Z is expressed as \[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
We are given that \[Z = \dfrac{{{{\left( {\sqrt 3 + i} \right)}^3}{{\left( {3i + 4} \right)}^2}}}{{{{\left( {8 + 6i} \right)}^2}}}\]
Therefore modulus of Z \[\left| Z \right| = \left| {\dfrac{{{{\left( {\sqrt 3 + i} \right)}^3}{{\left( {3i + 4} \right)}^2}}}{{{{\left( {8 + 6i} \right)}^2}}}} \right|\]
\[ = {\dfrac{{{{\left[ {\sqrt {{{\left( {\sqrt 3 } \right)}^2} + {1^2}} } \right]}^3}\left[ {\sqrt {{3^2} + {4^2}} } \right]}}{{{{\left[ {\sqrt {{8^2} + {6^2}} } \right]}^2}}}^2}\]
Which on simplification gives us the following expression
\[ = \dfrac{{{2^3}{{.5}^2}}}{{{{10}^2}}} = 2\]
Therefore we get \[\left| Z \right| = 2\]

Note: Complex numbers are the most wide field of sets of numbers. It comprises of all kind of number sets like natural numbers, real numbers,integers , rational and irrational numbers.The complex number comprises of two parts , real and imaginary parts and are written in the form of \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] (iota) is a symbol called the imaginary unit with value i2 = −1.