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Chapter 2: Problem 62

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

Short Answer

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

Step by step solution

01

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 \( -5 + 12i \).
02

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 \(\frac{-25i + 60}{(-5 + 12i)(-5 - 12i)}\).
03

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 \(\frac{-25i + 60}{169}\).
04

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 \(\frac{-25i}{169}+ \frac{60}{169} = \frac{60}{169} - \frac{25i}{169}\).

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Most popular questions from this chapter

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=\frac{3 x}{x-2}\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 6\)

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=3 x-3 \sqrt{x}-4$$

Operations with Rational Expressions Simplify the expression. $$\frac{2}{z+2}-\left(3-\frac{2}{z}\right)$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}+12 x+9 \leq 0$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|2 x-5|=11$$
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