Question 12 of 28beginner🔧 ApplyNumerical1 mark

Find the multiplicative inverse of the following complex number: 5+3i\sqrt{5} + 3i

Correct Answer

514−314\frac{\sqrt{5}}{14} - \frac{3}{14} i

Exercise: EXERCISE 4.1 | Q: 12 | (Page 8)
For More Understanding

Explanation

The multiplicative inverse of a complex number z = a + ib is given by z−1z^{-1} = (a - ib)/(a2a^{2} + b2b^{2}) = zˉ\bar{z}/|z|^2, as shown in Example 5 and the general formula in the textbook. For z = 5\sqrt{5} + 3i, a = 5\sqrt{5}, b = 3, so |z|^2 = 5 + 9 = 14, and zˉ=5\bar{z} = \sqrt{5} - 3i. Hence, the inverse is (5\sqrt{5} - 3i)/14.

Solution Steps

  1. Step 1: Identify a and b from the complex number: a = 5\sqrt{5}, b = 3.

  2. Step 2: Compute |z|^2 = a2a^{2} + b2b^{2} = (5\sqrt{5})^2 + 323^{2} = 5 + 9 = 14.

  3. Step 3: Write the conjugate zˉ=a\bar{z} = a - ib = 5\sqrt{5} - 3i.

  4. Step 4: Apply the formula: z−1z^{-1} = zˉ\bar{z} / |z|^2 = (5\sqrt{5} - 3i) / 14.

  5. Step 5: Express in standard form: 514−314\frac{\sqrt{5}}{14} - \frac{3}{14} i.

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