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Chapter 1: Problem 25

Divide and express the result in standard form. $$ \frac{8 i}{4-3 i} $$

Short Answer

Expert verified
The result of the division in standard form is \(-\frac{24}{25} + \frac{32}{25}i\).

Step by step solution

01

Identify and Multiply by the Conjugate

The conjugate of \(4 - 3i\) is \(4 + 3i\). Multiply the numerator and the denominator of the fraction by this conjugate:\[ \frac{8i * (4 + 3i)}{(4 - 3i) * (4 + 3i)}\]
02

Simplify the Numerator and Denominator

Simplify the multiplication you performed in the last step. You can distribute the factor in the numerator. For the denominator difference of squares formula can be used which states that for any real numbers a and b \((a - b)(a + b) = a^2 - b^2\):Numerator simplification: \[8i * (4 + 3i) = 32i + 24i^2\]Remember that \(i^2 = -1\), so you can replace \(24i^2\) with \(-24\). Hence the numerator simplifies to \(32i -24\).Denominator simplification using the formula (\(a^2 - b^2\)):\[(4 - 3i) * (4 + 3i) = 4^2 - (3i)^2 = 16 - (9 * -1) = 16 + 9 = 25\]Hence, the fraction simplifies to:\[\frac{32i -24}{25}\]
03

Divide the Numerator by the Denominator

It's best to write complex numbers in the form a + bi. Divide both the real and imaginary parts of the numerator by the denominator to achieve this:\[-\frac{24}{25} + \frac{32}{25}i\]

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Most popular questions from this chapter

Explaining the Concepts. Describe how to solve an absolute value inequality involving the symbol \(<\). Give an example.

When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

Solve equation by the method of your choice. $$ \sqrt{2} x^{2}+3 x-2 \sqrt{2}=0 $$

Use the Pythagorean Theorem and the square root property to solve. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 4 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally across the park?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.
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