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Q10:
Express the complex number 
in the form $a + ib$.
Express the complex number in the form $a + ib$.
Solution :
Step 1: Identifying the Given Expression
The expression provided is $z = \left( -2 - \frac{1}{3}i \right)^3$. We aim to express this in the standard algebraic form $z = a + ib$, where $a, b \in \mathbb{R}$.
Step 2: Applying the Binomial Identity
To expand a cube of the form $(x + y)^3$, we use the algebraic identity [Binomial Theorem for $n=3$]:
$(x + y)^3 = x^3 + y^3 + 3xy(x + y)$
Substituting $x = -2$ and $y = -\frac{1}{3}i$:
$z = (-2)^3 + \left( -\frac{1}{3}i \right)^3 + 3(-2)\left( -\frac{1}{3}i \right) \left( -2 - \frac{1}{3}i \right)$
Step 3: Calculating Individual Terms
- $(-2)^3 = -8$
- $\left( -\frac{1}{3}i \right)^3 = -\frac{1}{27}i^3$
- $3(-2)\left( -\frac{1}{3}i \right) = 2i$
Since $i^2 = -1$, it follows that $i^3 = i^2 \cdot i = -i$. Therefore, $-\frac{1}{27}(-i) = \frac{1}{27}i$.
Step 4: Algebraic Expansion and Substitution
Now substitute these values back into the expanded equation:
$z = -8 + \frac{1}{27}i + 2i(-2 - \frac{1}{3}i)$
Distribute $2i$ into the binomial expression:
$z = -8 + \frac{1}{27}i + (2i \cdot -2) + (2i \cdot -\frac{1}{3}i)$
$z = -8 + \frac{1}{27}i - 4i - \frac{2}{3}i^2$
Step 5: Simplifying using the Imaginary Unit Property
Applying the property $i^2 = -1$ [Fundamental Theorem of Complex Numbers]:
$z = -8 + \frac{1}{27}i - 4i - \frac{2}{3}(-1)$
$z = -8 + \frac{1}{27}i - 4i + \frac{2}{3}$
Step 6: Real and Imaginary Part Grouping
Group the real parts and the imaginary parts separately:
Real part ($a$): $-8 + \frac{2}{3} = \frac{-24 + 2}{3} = -\frac{22}{3}$
Imaginary part ($b$): $\left( \frac{1}{27} - 4 \right)i = \left( \frac{1 - 108}{27} \right)i = -\frac{107}{27}i$
Final Solution: -\frac{22}{3} - \frac{107}{27}i
More Questions from Class 11 Mathematics Complex Numbers EXERCISE 4.1
- Q1: Express the complex number in the form $a + ib$.
- Q2: Express the complex number $i^9 + i^{19}$ in the form $a + ib$.
- Q3: Express the complex number $i^{-39}$ in the form $a + ib$.
- Q4: Express the complex number $3(7 + i7) + i(7 + i7)$ in the form $a + ib$.
- Q5: Express the complex number $(1 - i) - (-1 + i6)$ in the form $a + ib$.
- Q6: Express the complex number in the form $a + ib$.
- Q7: Express the complex number in the form $a + ib$.
- Q8: Express the complex number $(1 - i)^4$ in the form $a + ib$.
- Q9: Express the complex number in the form $a + ib$.
- Q11: Find the multiplicative inverse of the complex number $4 - 3i$.
- Q12: Find the multiplicative inverse of the complex number .
- Q13: Find the multiplicative inverse of the complex number $-i$.
- Q14: Express the following expression in the form of $a + ib$:
CBSE Solutions for Class 11 Mathematics Complex Numbers
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Complex Numbers
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With years of experience teaching Class 11 Mathematics, I have honed my skills to provide students with a deep understanding of complex concepts. My teaching approach is student-centered, focusing on clarity, problem-solving techniques, and real-world applications. I have a strong command over the subject, making intricate topics accessible and engaging. My classes are interactive and designed to foster a genuine interest in mathematics. By incorporating innovative methods and advanced tools, I ensure that learning is both effective and enjoyable. Join my classes to experience a supportive learning environment where your academic success is the ultimate goal!
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