Answer

Verified

394.2k+ views

**Hint:**In this question, we have to simplify the given expression. First we need to break the square of the expression into multiplication of the same term twice. Then we need to multiply terms with each other. On simplifying that we will get the required solution.

**Complete step-by-step solution:**

It is given that, \[{\left( {5 - 3i} \right)^2}\]

We need to simplify the given expression \[{\left( {5 - 3i} \right)^2}\]

To simplify the given expression we first need to break the square of the expression into multiplication of the same term twice. Then we need to multiply terms with each other.

\[ \Rightarrow {\left( {5 - 3i} \right)^2} = \left( {5 - 3i} \right)\left( {5 - 3i} \right)\]

Let us multiply with the first and second term and we get

\[ \Rightarrow {\left( {5 - 3i} \right)^2} = 5\left( {5 - 3i} \right) - 3i\left( {5 - 3i} \right)\]

On multiply the term and we get

\[ \Rightarrow 5 \times 5 - 5 \times 3i - 3i \times 5 + \left( { - 3i} \right) \times \left( { - 3i} \right)\]

Thus we get

\[ \Rightarrow 25 - 15i - 15i + 9{i^2}\]

On add the term and we get

\[ \Rightarrow 25 - 30i + 9{i^2}\]

We know that the value of \[{i^2} = - 1\] where i is an imaginary number]

\[ \Rightarrow 25 - 30i - 9\]

Thus we get

\[ \Rightarrow 16 - 30i\]

Therefore, \[{\left( {5 - 3i} \right)^2} = 16 - 30i\]

**Hence simplifying \[{\left( {5 - 3i} \right)^2}\] we get \[16 - 30i\]**

**Additional information:**

The expression contains i and we need to know what is \[i\] .

A complex number is a number that can be expressed in the form \[a + bi\] where a and b are real numbers and \[i\] represents the imaginary unit, satisfying the equation \[{i^2} = - 1\] .Since no real number satisfies this equation , \[i\] is called an imaginary number.

**Note:**There is an alternative method as follows:

It is given that, \[{\left( {5 - 3i} \right)^2}\] .

We need to simplify \[{\left( {5 - 3i} \right)^2}\] .

To simplify the given expression we first need to apply one algebraic formula \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\] to simplify it. After applying the formula we will put the value of \[{i^2}\] which is equal to \[ - 1\] .

\[{\left( {5 - 3i} \right)^2}\]

\[ = {5^2} - 2 \times 5 \times 3i + {\left( { - 3i} \right)^2}\] [Applying \[a = 5\& b = 3i\] in the formula \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\] ]

Simplifying we get,

\[ = 25 - 30i + 9{i^2}\]

\[ = 25 - 30i - 9\] [we know that the value of \[{i^2} = - 1\] where \[i\] is an imaginary number]

Again simplifying we get,

\[ = 16 - 30i\]

Therefore, \[{\left( {5 - 3i} \right)^2} = 16 - 30i\] .

Hence simplifying \[{\left( {5 - 3i} \right)^2}\] we get \[16 - 30i\] .

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE