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Chapter 2: Problem 81

Perform the indicated operations and write the result in standard form. $$\frac{4}{(2+i)(3-i)}$$

Short Answer

Expert verified
The result of the operation is \( \frac{1}{6} + 0i \)

Step by step solution

01

Determine the Conjugate

Firstly, determine the conjugate of each binomial in the denominator, which is the binomial with the sign of its imaginary term switched. Conjugate of \(2+i\) is \(2-i\) and conjugate of \(3-i\) is \(3+i\).
02

Simplify the Denominator

Multiply the denominators with their respective conjugates and simplify it: \[(2+i)(2-i) = 2(2) + 2(-i) + i(2) + i(-i) = 4 - 2i + 2i - 1 = 3\]Also, you get \[(3-i)(3+i) = 3(3) - 3(i) + i(3) - i(-i) = 9 - 3i + 3i - 1 = 8\]Hence, the denominator becomes 3*8 which equals to 24.
03

Simplify the Expression

Finally, simplify the whole expression where the numerator remains the same, and the denominator is now a real number 24:\[\frac{4}{24} = \frac{1}{6}\]
04

Write in Standard Form

The standard form for a complex number is a + bi, where a and b are real numbers. In this case, the result is a real number, hence the standard form would be \( \frac{1}{6} + 0i \).

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Most popular questions from this chapter

Solve each inequality using a graphing utility. $$\frac{x+2}{x-3} \leq 2$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2} \leq 0$$

Describe how to graph a rational function.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+2}-2$$
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