%2F(3-2i).jpeg "(8-i)/(3-2i)")
Simplifying Complex Fractions: (8-i)/(3-2i)
In mathematics, complex numbers often appear in fractions, requiring simplification. This article explores the process of simplifying the fraction (8-i)/(3-2i).
Understanding Complex Fractions
A complex fraction is a fraction where either the numerator, denominator, or both contain complex numbers. Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1.
Simplifying (8-i)/(3-2i)
To simplify this fraction, we employ the technique of multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number (a + bi) is (a - bi).
Step 1: Find the Conjugate
The conjugate of (3 - 2i) is (3 + 2i).
Step 2: Multiply Numerator and Denominator
Multiply both the numerator and denominator of the fraction by the conjugate:
[(8 - i) * (3 + 2i)] / [(3 - 2i) * (3 + 2i)]
Step 3: Expand and Simplify
Expand the products in the numerator and denominator:
[(8 * 3) + (8 * 2i) - (i * 3) - (i * 2i)] / [(3 * 3) + (3 * 2i) - (2i * 3) - (2i * 2i)]
Simplify the expressions:
[24 + 16i - 3i + 2] / [9 + 6i - 6i + 4]
Combine real and imaginary terms:
[26 + 13i] / [13]
Step 4: Final Simplification
Divide both the numerator and denominator by 13:
(2 + i)
Therefore, the simplified form of (8-i)/(3-2i) is 2 + i.
Conclusion
Simplifying complex fractions often involves multiplying by the conjugate of the denominator. This process eliminates the imaginary term in the denominator, resulting in a simpler expression. In the case of (8-i)/(3-2i), we successfully simplified it to the complex number 2 + i.