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Chapter 2: Problem 31

Perform the operation and write the result in standard form. $$-\left(\frac{3}{2}+\frac{5}{2} i\right)+\left(\frac{5}{3}+\frac{11}{3} i\right)$$

Short Answer

Expert verified
-\frac{1}{6} + \frac{7}{6}i

Step by step solution

01

Distribute Negative Sign

First distribute the negative sign in the first complex number to both the real and imaginary parts. This gives us: \(-\frac{3}{2} - \frac{5}{2}i\)
02

Add Real Parts

Next, add the real parts (numbers without \(i\)). This means adding \(-\frac{3}{2}\) and \(\frac{5}{3}\). Which gives us: \(-\frac{3}{2} + \frac{5}{3} = -\frac{1}{6}\)
03

Add Imaginary Parts

Now, add the imaginary parts (numbers with \(i\)). This means adding \(-\frac{5}{2}i\) and \(\frac{11}{3}i\). Which gives us: \(-\frac{5}{2}i + \frac{11}{3}i = \frac{7}{6}i\)
04

Combine Real and Imaginary Parts

Now we combine the results from steps 2 and 3 to write in standard form. The real part is \(-\frac{1}{6}\) and the imaginary part is \(\frac{7}{6}i\). Therefore, the answer is: \(-\frac{1}{6} + \frac{7}{6}i\)

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Most popular questions from this chapter

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Decide whether the statement is true or false. Justify your answer. If \(x=-i\) is a zero of the function \(f(x)=x^{3}+i x^{2}+i x-1\) then \(x=i\) must also be a zero of \(f\)

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b)\) (GRAPH CANNOT COPY) Match each complex number with its corresponding point. (i) 3 (ii) \(3 i\) (iii) \(4+2 i\) (iv) \(2-2 i\) (v) \(-3+3 i\) (vi) \(-1-4 i\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.
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