Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the quotient in standard form. $$\frac{5 i}{(2+3 i)^{2}}$$

Step-by-Step Solution

Answer

The quotient in standard form is \( \frac{60}{169} - \frac{25i}{169} \).

Step 1: Expand the Denominator

First, expand the denominator which is \( (2+3i)^2 \). The operation follows the square of a binomial pattern \((a + b)^2 = a^2 + 2ab + b^2\), giving us \(2^2+ 2*2*3i+ (3i)^2 = 4 + 12i - 9 \), because \(i^2 = -1\). Thus, the denominator becomes \( -5 + 12i \).

Step 2: Simplify the Complex Fraction

Now, simplify the complex fraction. A standard technique for this is to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number \(a+bi\) is \(a-bi\). So here it is \( -5 - 12i \). Multiply both the numerator and denominator by \( -5 - 12i \): \(\frac{5i(-5 - 12i)}{(-5 + 12i)(-5 - 12i)}\). This simplifies to \(\frac{-25i - 60i^2}{(-5 + 12i)(-5 - 12i)}\). Since \(i^2 = -1\), so the equation further simplifies to \(\frac{-25i + 60}{(-5 + 12i)(-5 - 12i)}\).

Step 3: Multiply the Conjugates in the Denominator

In this step, we multiply the conjugates in the denominator. This always results in a real number. \( (-5 + 12i)(-5 - 12i) = (-5)^2 - (12i)^2 = 25 - (-144) = 169 \). So our simplified fraction so far is \(\frac{-25i + 60}{169}\).

Step 4: Separate into Real and Imaginary Parts

The last step is to rewrite the fraction we obtained in standard form, i.e., separately in terms of the real and imaginary parts. This gives us \(\frac{-25i}{169}+ \frac{60}{169} = \frac{60}{169} - \frac{25i}{169}\).