Question

How do you simplify \[5i.i - 2i\]?

Answer 1

Hint: In order to simplify the above complex number, we have to first combine the $ i.i $ part by using the product property of exponent which is $ {a^m}.{a^n} = {a^{m + n}} $ .Replacing the $ {i^2} $ with $ - 1 $ ,your will get your required simplification .
 $ {i^2} = - 1 $
 $ {a^m}.{a^n} = {a^{m + n}} $

Complete step-by-step answer:
We are Given a complex number\[5i.i - 2i\]let it be z
 $ z = 5i.i - 2i $ --(1)
Here i is the imaginary number which is commonly known as the iota.
In order to simplify the given complex number, we will combine the $ i.i $ part by using the product property of exponents which states that when we have a product of two numbers having the same base but different exponent value then the base stays the same and we add both the exponents.
If $ a $ is a real number and $ m\,and\,n $ are integers, then $ {a^m}.{a^n} = {a^{m + n}} $
Applying this rule to the equation (1) , we get
 $
  z = 5{i^{1 + 1}} - 2i \\
  z = 5{i^2} - 2i \;
  $
And as we know that the value of $ {i^2} $ is equal to $ - 1 $ ,so replacing it in the above equation
 $
  z = 5\left( { - 1} \right) - 2i \\
  z = - 5 - 2i \;
  $
Therefore, the simplification of complex numbers \[5i.i - 2i\]is equal to $ - 5 - 2i $ .
So, the correct answer is “ $ - 5 - 2i $ ”.

Note: 1. Real Number: Any number which is available in a number system, for example, positive, negative, zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance: 12, - 45, 0, 1/7, 2.8, √5, and so forth, are all the real numbers.
2. A Complex number is a number which are expressed in the form $ a + ib $ where $ ib $ is the imaginary part and $ a $ is the real number .i is generally known by the name iota. \[\]
or in simple words complex numbers are the combination of a real number and an imaginary number .
3.The Addition or multiplication of any 2-conjugate complex number always gives an answer which is a real number.
1. Complex numbers are very useful in representing periodic motion like water waves, light waves and current and many more things which depend on sine or cosine waves.
2. Complex conjugate of $ a + ib $ is $ a - ib $
3. $ {i^3} $ is equal to $ - i $ as $ {i^3} = i.{i^2} = i.\left( { - 1} \right) = - i $