-((3%2B4i)(4%2B5i))%2F((4%2B3i)(6%2B7i)).jpeg "(iv) ((3+4i)(4+5i))/((4+3i)(6+7i))")
Simplifying Complex Fractions: A Step-by-Step Guide
This article will guide you through simplifying the complex fraction:
((3 + 4i)(4 + 5i)) / ((4 + 3i)(6 + 7i))
1. Expanding the Numerator and Denominator
First, we'll expand both the numerator and the denominator using the FOIL method (First, Outer, Inner, Last):
Numerator: (3 + 4i)(4 + 5i) = (3 * 4) + (3 * 5i) + (4i * 4) + (4i * 5i) = 12 + 15i + 16i + 20i²
Denominator: (4 + 3i)(6 + 7i) = (4 * 6) + (4 * 7i) + (3i * 6) + (3i * 7i) = 24 + 28i + 18i + 21i²
2. Simplifying using i² = -1
Recall that i² = -1. Substitute this value into both the numerator and denominator:
Numerator: 12 + 15i + 16i + 20i² = 12 + 15i + 16i - 20 = -8 + 31i
Denominator: 24 + 28i + 18i + 21i² = 24 + 28i + 18i - 21 = 3 + 46i
3. Multiplying by the Conjugate of the Denominator
To eliminate the imaginary part in the denominator, we'll multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 3 + 46i is 3 - 46i.
((3 + 4i)(4 + 5i)) / ((4 + 3i)(6 + 7i)) = ((-8 + 31i) * (3 - 46i)) / ((3 + 46i) * (3 - 46i))
4. Expanding and Simplifying
Again, using the FOIL method:
Numerator: (-8 + 31i)(3 - 46i) = (-8 * 3) + (-8 * -46i) + (31i * 3) + (31i * -46i) = -24 + 368i + 93i - 1426i² = -24 + 368i + 93i + 1426 = 1402 + 461i
Denominator: (3 + 46i)(3 - 46i) = (3 * 3) + (3 * -46i) + (46i * 3) + (46i * -46i) = 9 - 138i + 138i - 2116i² = 9 + 2116 = 2125
5. Final Simplification
Now we have:
((3 + 4i)(4 + 5i)) / ((4 + 3i)(6 + 7i)) = (1402 + 461i) / 2125
This can be expressed in the standard form of a complex number:
((3 + 4i)(4 + 5i)) / ((4 + 3i)(6 + 7i)) = (1402/2125) + (461/2125)i
Therefore, the simplified form of the given complex fraction is (1402/2125) + (461/2125)i.