Express the given complex number in the form\( a + ib: (-2-\dfrac{1}{3}i)^3\)
Solution and Explanation
\((-2-\dfrac{1}{3}i)^3\)
\(=-[2^3+(\dfrac{i}{3})^3+3.(2)(\dfrac{i}{3})(2)(2+\dfrac{i}{3})]\)
\(=-[8+\dfrac{i^3}{27}+2i(2+\dfrac{i}{3})]\)
\(=-[8-\dfrac{i}{27}+4i-\dfrac{2}{3}]\)
\(=-[\dfrac{22}{7}+\dfrac{107}{27}i]\)
\(=-\dfrac{22}{3}-\dfrac{107}{27}i\) (Ans.)
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
