5. Plot each complex number and find its absolute value a) \( 3+4 i \) b) -4 c) \( -5 i \) d) \( 3-2 i \) e) \( \frac{1}{(2+i)(\sqrt{3}-2 i)} \) f) \( 5-4 i+\frac{5}{3-4 i} \)

Let’s crank up the fun as we explore these complex numbers! First, let’s plot each number on the complex plane, where the x-axis represents the real part and the y-axis the imaginary part. a) \( 3 + 4i \) is plotted at (3, 4). Its absolute value is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \), which gives \( |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \). b) The number -4 is plotted at (-4, 0). Its absolute value is simply \( |-4| = 4 \). c) \( -5i \) is plotted at (0, -5). Its absolute value is \( |-5i| = 5 \). d) \( 3 - 2i \) is plotted at (3, -2). Its absolute value is \( |3 - 2i| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \). e) For \( \frac{1}{(2+i)(\sqrt{3}-2i)} \), first simplify the denominator: \( (2+i)(\sqrt{3}-2i) = 2\sqrt{3} - 4i + i\sqrt{3} + 2 = 2\sqrt{3} + 2 + (i\sqrt{3} - 4i) = 2 + 2\sqrt{3} + i(\sqrt{3}-4) \). Finding the absolute value of this whole complex expression might take a bit, but it’s an adventure in simplifying! f) \( 5 - 4i + \frac{5}{3 - 4i} \) requires some careful calculation. First, multiply the numerator and denominator of the fraction by the conjugate of the denominator to simplify. After that, add the components to \( 5 - 4i \) to produce a result—don’t forget to plot it! The absolute value here can be tricky and may entail a little extra arithmetic, but that’s where the real fun begins. With these steps, you can not only plot and find absolute values but also appreciate the beauty and complexity of numbers!