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Chapter 1: Problem 75

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

Short Answer

Expert verified
The simplified form of the given complex fraction is \(1.75 - 0.5i\).

Step by step solution

01

Identify the Conjugate of the Denominator

The conjugate of the denominator, which is \((2 + i)(3 - i)\), is \((2 - i)(3 + i)\). The conjugate of any complex number \((a + bi)\) is \((a - bi)\).
02

Multiply the Numerator and Denominator by the Conjugate of the Denominator

To simplify the complex fraction, multiply both the numerator and the denominator by the conjugate of the denominator, \((2 - i)(3 + i)\). This gives \(\frac{4(2 - i)(3 + i)}{(2 + i)(3 - i)(2 - i)(3 + i)}\).
03

Simplify the Numerator and the Denominator Separately

Expand the expression in the numerator and the denominators separately.\nThe numerator expands to \(4(6 + i - 3i - i^2)\) = \(4(6 - 2i + 1)\) = \(4(7 - 2i)\), which simplifies to \(28 - 8i\). \nThe denominator expands to \(6 - 2i + 3i + i^2 - 2i + 4 + 2i^2 - 3i + 2i + 4 - i^2\) = \(14 + 0i - 2i^2 = 14 + 2 = 16\).
04

Write the Final Answer in Standard Form

Finally, divide the numerator by the denominator to find the quotient: \(\frac{28 - 8i}{16}\) = \(1.75 - 0.5i\). This is the final answer in the standard form.

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Most popular questions from this chapter

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 75 cubic inches, find the length and width of the open box.

Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6} $$

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$V=P r t \text { for } P$$
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