Question 10 of 28intermediate🔧 ApplyNumerical2 marks

Express the following complex number in the form a+iba + ib: (−2−13i)3\left(-2 - \frac{1}{3}i\right)^3

Correct Answer

Step 1: Expand using binomial theorem: (−2−13i)3=(−2)3+3(−2)2(−13i)+3(−2)(−13i)2+(−13i)3.\left(-2 - \frac{1}{3}i\right)^3 = (-2)^3 + 3(-2)^2\left(-\frac{1}{3}i\right) + 3(-2)\left(-\frac{1}{3}i\right)^2 + \left(-\frac{1}{3}i\right)^3.

Step 2: Compute each term:

  • (−2)3=−8(-2)^3 = -8
  • 3(−2)2(−13i)=3×4×(−13i)=−4i3(-2)^2\left(-\frac{1}{3}i\right) = 3 \times 4 \times \left(-\frac{1}{3}i\right) = -4i
  • 3(−2)(−13i)2=3(−2)(19i2)=3(−2)(−19)=233(-2)\left(-\frac{1}{3}i\right)^2 = 3(-2)\left(\frac{1}{9}i^2\right) = 3(-2)\left(-\frac{1}{9}\right) = \frac{2}{3}
  • (−13i)3=(−13)3i3=−127⋅(−i)=127i\left(-\frac{1}{3}i\right)^3 = \left(-\frac{1}{3}\right)^3 i^3 = -\frac{1}{27} \cdot (-i) = \frac{1}{27}i

Step 3: Combine real and imaginary parts: Real part: −8+23=−223-8 + \frac{2}{3} = -\frac{22}{3} Imaginary part: −4i+127i=−10727i-4i + \frac{1}{27}i = -\frac{107}{27}i

Step 4: Write in the form a+iba+ib:

(−2−13i)3=−223−10727i.\left(-2 - \frac{1}{3}i\right)^3 = -\frac{22}{3} - \frac{107}{27}i.
Exercise: EXERCISE 4.1 | Q: 10 | (Page 8)
For More Understanding

Explanation

This question is similar to Example 3 in the textbook, where we expand (a - b)^3 using the binomial theorem and simplify using i2i^{2} = -1 and i3i^{3} = -i. The steps involve expanding the cube, computing each term, and then combining real and imaginary parts to express the result in the standard form a + ib.

Solution Steps

  1. Expand using binomial theorem: (x+y)^3 = x3x^{3} + 3x23x^{2} y + 3x y2y^{2} + y3y^{3}, with x = -2, y = -i/3.

  2. Compute each term: x3x^{3} = -8; 3x23x^{2} y = -4i; 3x y2y^{2} = 2/3; y3y^{3} = (1/27)i.

  3. Simplify using i2i^{2} = -1 and i3i^{3} = -i.

  4. Combine real parts: -8 + 2/3 = -22/3.

  5. Combine imaginary parts: -4i + (1/27)i = -107/27 i.

  6. Write final answer: -22/3 - (107/27)i.

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