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Magazine Chapter 7: Problem 121
Simplify. $$\frac{6}{1+\frac{3}{i}}$$
Short Answer
Expert verified
\( \frac{3}{5} + \frac{9i}{5} \)
Step by step solution
01
Identify the expression inside the denominator
The given expression is \(\frac{6}{1+\frac{3}{i}}\). First, focus on simplifying the denominator, which is \(1 + \frac{3}{i}\).
02
Simplify the fraction in the denominator
To eliminate the fraction inside the denominator, multiply the numerator and the denominator inside the denominator by \i\: \( \frac{3}{i} \cdot\ \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i \).
03
Rewrite the denominator
Substitute \-3i\ for \frac{3}{i}\ back into the denominator: \(1 - 3i\). So the expression now is \(\frac{6}{1 - 3i}\).
04
Rationalize the denominator
To remove the imaginary number from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator: \(1 + 3i\). This turns the expression into \(\frac{6(1 + 3i)}{(1 - 3i)(1 + 3i)}\).
05
Simplify the numerator
Expand the numerator: \(6 \cdot\ (1 + 3i) = 6 + 18i\).
06
Simplify the denominator
Expand and simplify the denominator using the difference of squares: \( (1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10 \).
07
Combine results
Combine the results from the numerator and the denominator: \( \frac{6 + 18i}{10} = \frac{6}{10} + \frac{18i}{10} = \frac{3}{5} + \frac{9i}{5} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Numbers
Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary unit is denoted by \(i\), where \(i = \sr{-1}\). For example, in the expression \(2 + 3i\), the number 2 is the real part, and \(3i\) is the imaginary part.
Key points about complex numbers:
Key points about complex numbers:
- The real part is a real number.
- The imaginary part is a real number multiplied by \(i\).
- Complex numbers follow the same arithmetic rules as real numbers, but including \(i\). For instance, \(i^2 = -1\).
Rationalizing the Denominator
Rationalizing the denominator means eliminating the imaginary part from the denominator of a fraction. This is done by multiplying the numerator and denominator by the conjugate of the denominator.
Steps to rationalize the denominator:
Steps to rationalize the denominator:
- Identify the complex conjugate: for any complex number \(a + bi\), its conjugate is \(a - bi\).
- Multiply both the numerator and the denominator by the conjugate.
- Use the formula for the difference of squares: \((a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2\) since \(i^2 = -1\).
Fractions
Fractions in mathematics represent parts of a whole. Simplifying fractions means making them as simple as possible. In our exercise, we simplify a complex fraction:
\text{1} The outer fraction is \(\frac{6}{1 + \frac{3}{i}}\). To simplify, follow these steps:
Result: By following these steps, we reduce \(\frac{6}{1 + \frac{3}{i}}\) to \(\frac{3}{5} + \frac{9i}{5}\).
\text{1} The outer fraction is \(\frac{6}{1 + \frac{3}{i}}\). To simplify, follow these steps:
- First, simplify any fractions within the numerator or the denominator.
- Next, rationalize the denominator if it includes complex numbers.
- After that, combine the results if possible.
Result: By following these steps, we reduce \(\frac{6}{1 + \frac{3}{i}}\) to \(\frac{3}{5} + \frac{9i}{5}\).
