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# How do you simplify ${\left( {5 - 3i} \right)^2}$ ?

Last updated date: 19th Jul 2024
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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 $16 - 30i$

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 $16 - 30i$ .