
How do you simplify: $\dfrac{{2 + 3i}}{{3 - 2i}}$ ?
Answer
445.5k+ views
Hint:In the given problem, we are required to simplify an expression involving complex numbers. For simplifying the given expression, we need to have a thorough knowledge of complex number sets and its applications in such questions. Algebraic rules and properties also play a significant role in simplification of such expressions.
Complete step by step solution:
In the question, we are given an expression which needs to be simplified using the knowledge of complex number sets.
For simplifying the given expression involving complex numbers, we need to first multiply the numerator and denominator with the conjugate of the complex number present in the denominator so as to obtain a real number in the denominator.
So, $\dfrac{{2 + 3i}}{{3 - 2i}} = \dfrac{{2 + 3i}}{{3 - 2i}} \times \left( {\dfrac{{3 + 2i}}{{3 + 2i}}} \right)$
Using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$,
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{{{\left( 3 \right)}^2} - {{\left( {2i} \right)}^2}}}\]
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{9 - 4{i^2}}}\]
We know that ${i^2} = - 1$. Hence, substituting ${i^2}$ as $ - 1$, we get,
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{9 - 4\left( { - 1} \right)}}\]
Opening brackets and simplifying further, we get,
\[ = \dfrac{{2\left( {3 + 2i} \right) + 3i\left( {3 + 2i} \right)}}{{9 + 4}}\]
\[ = \dfrac{{\left( {6 + 4i} \right) + \left( {9i + 6{i^2}} \right)}}{{9 + 4}}\]
Distributing the denominator to both the terms, we get,
\[ = \dfrac{{6 + 4i + 9i - 6}}{{9 + 4}}\]
\[ = \dfrac{{13i}}{{13}}\]
\[ = i\]
Therefore, the given expression $\dfrac{{2 + 3i}}{{3 - 2i}}$ can be simplified as: \[i\] .
Note: The given problem revolves around the application of properties of complex numbers in questions. The question tells us about the wide ranging significance of the complex number set and its properties. The final answer can also be verified by working the solution backwards and getting back the given expression $\dfrac{{2 + 3i}}{{3 - 2i}}$. Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number sets and its applications in such questions.
Complete step by step solution:
In the question, we are given an expression which needs to be simplified using the knowledge of complex number sets.
For simplifying the given expression involving complex numbers, we need to first multiply the numerator and denominator with the conjugate of the complex number present in the denominator so as to obtain a real number in the denominator.
So, $\dfrac{{2 + 3i}}{{3 - 2i}} = \dfrac{{2 + 3i}}{{3 - 2i}} \times \left( {\dfrac{{3 + 2i}}{{3 + 2i}}} \right)$
Using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$,
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{{{\left( 3 \right)}^2} - {{\left( {2i} \right)}^2}}}\]
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{9 - 4{i^2}}}\]
We know that ${i^2} = - 1$. Hence, substituting ${i^2}$ as $ - 1$, we get,
\[ = \dfrac{{\left( {2 + 3i} \right)\left( {3 + 2i} \right)}}{{9 - 4\left( { - 1} \right)}}\]
Opening brackets and simplifying further, we get,
\[ = \dfrac{{2\left( {3 + 2i} \right) + 3i\left( {3 + 2i} \right)}}{{9 + 4}}\]
\[ = \dfrac{{\left( {6 + 4i} \right) + \left( {9i + 6{i^2}} \right)}}{{9 + 4}}\]
Distributing the denominator to both the terms, we get,
\[ = \dfrac{{6 + 4i + 9i - 6}}{{9 + 4}}\]
\[ = \dfrac{{13i}}{{13}}\]
\[ = i\]
Therefore, the given expression $\dfrac{{2 + 3i}}{{3 - 2i}}$ can be simplified as: \[i\] .
Note: The given problem revolves around the application of properties of complex numbers in questions. The question tells us about the wide ranging significance of the complex number set and its properties. The final answer can also be verified by working the solution backwards and getting back the given expression $\dfrac{{2 + 3i}}{{3 - 2i}}$. Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number sets and its applications in such questions.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Trending doubts
10 examples of friction in our daily life

Difference Between Prokaryotic Cells and Eukaryotic Cells

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

Define least count of vernier callipers How do you class 11 physics CBSE
