Simplifying Complex Fractions: How to Simplify 2i / (3 - i√2)

When dealing with complex fractions, it's essential to understand various techniques such as rationalization to simplify the expression. In this guide, we will demonstrate how to simplify the given complex fraction 2i / (3 - i√2). This process involves rationalizing the denominator and simplifying the expression. Follow along to grasp the step-by-step process of simplifying complex fractions efficiently.

Understanding the Problem

The problem given is 2i / (3 - i√2), where 'i' is the imaginary unit with the property that (i^2 -1).

Rationalizing the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of (3 - i√2) is (3 i√2).

(frac{2i}{3 - isqrt{2}} cdot frac{3 isqrt{2}}{3 isqrt{2}})

This multiplication helps to eliminate the imaginary part from the denominator.

Intermediate Steps

Multiply the numerator and the denominator: Numerator: (2i cdot (3 isqrt{2}) 6i 2i^2sqrt{2}) Denominator: ((3 - isqrt{2}) cdot (3 isqrt{2}) 9 2i^2)

Simplifying the Expression

Now, simplify each part of the expression:

Numerator: (6i 2(-1)sqrt{2} 6i - 2sqrt{2})

Denominator: (9 2(-1) 9 - 2 7)

Final Simplified Expression

The expression simplifies to:

(frac{6i - 2sqrt{2}}{7})

Further Simplification

We can separate the real and imaginary parts:

(frac{6i}{7} - frac{2sqrt{2}}{7})

Conclusion

Thus, the simplified form of the given complex fraction 2i / (3 - i√2) is:

(frac{6i - 2sqrt{2}}{7} frac{6i}{7} - frac{2sqrt{2}}{7})

Understanding and mastering these techniques are crucial in handling complex numbers and fractions in algebraic contexts. Whether you're a student, teacher, or a professional, the ability to simplify complex fractions is a valuable skill. Practice with similar problems to enhance your algebraic skills and intuition.