Question 14 of 28intermediate🔧 ApplyNumerical2 marks

Express the following expression in the form of a+iba + ib: (3+i5)(3−i5)(3+2i)−(3−i2)\frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})}

Correct Answer

−722i-\frac{7\sqrt{2}}{2}i or 0−722i0 - \frac{7\sqrt{2}}{2}i

Exercise: EXERCISE 4.1 | Q: 14 | (Page 8)
For More Understanding

Explanation

This question tests the concept of conjugate pairs and simplification of complex numbers. The numerator is a product of conjugates which simplifies to a real number using the identity (a+ib)(a-ib) = a² + b². The denominator involves subtraction of complex numbers, followed by rationalization to express the final answer in standard form a + ib.

Solution Steps

  1. Step 1: Simplify the numerator using the property of conjugates.

    (3+i5)(3−i5)=32+(5)2=9+5=14(3 + i\sqrt{5})(3 - i\sqrt{5}) = 3^2 + (\sqrt{5})^2 = 9 + 5 = 14

  2. Step 2: Simplify the denominator by subtracting the complex numbers.

    (3+2i)−(3−i2)=3+2i−3+i2=22i(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2}) = \sqrt{3} + \sqrt{2}i - \sqrt{3} + i\sqrt{2} = 2\sqrt{2}i

  3. Step 3: Write the expression with simplified numerator and denominator.

    1422i\frac{14}{2\sqrt{2}i}

  4. Step 4: Rationalize by multiplying numerator and denominator by ii.

    =1422i×ii=14i22⋅i2=14i22⋅(−1)=14i−22= \frac{14}{2\sqrt{2}i} \times \frac{i}{i} = \frac{14i}{2\sqrt{2} \cdot i^2} = \frac{14i}{2\sqrt{2} \cdot (-1)} = \frac{14i}{-2\sqrt{2}}

  5. Step 5: Simplify to get the final answer in the form a+iba + ib.

    =−7i2=−72i2=0−722i= \frac{-7i}{\sqrt{2}} = \frac{-7\sqrt{2}i}{2} = 0 - \frac{7\sqrt{2}}{2}i
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