Problem 46
Question
Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{6+2 i}{4-3 i} $$
Step-by-Step Solution
Verified Answer
The quotient in the form \( a + bi \) is \( \frac{18}{25} + \frac{26}{25}i \).
1Step 1: Identify the expression
We need to simplify the expression \( \frac{6+2i}{4-3i} \) into the form \( a + bi \), where \( a \) and \( b \) are real numbers.
2Step 2: Write the conjugate of the denominator
The conjugate of the denominator \( 4 - 3i \) is \( 4 + 3i \). We'll multiply the numerator and the denominator by this conjugate.
3Step 3: Multiply numerator and denominator by the conjugate
Multiply the numerator \( 6 + 2i \) and the denominator \( 4 - 3i \) by \( 4 + 3i \) to rationalize the denominator. \[ \frac{(6+2i)(4+3i)}{(4-3i)(4+3i)} \]
4Step 4: Expand the products
Apply the distributive property to expand the expression in the numerator and the denominator: Numerator: \((6)(4) + (6)(3i) + (2i)(4) + (2i)(3i) = 24 + 18i + 8i + 6i^2 \) Denominator:\((4)(4) + (4)(3i) - (3i)(4) - (3i)(3i) = 16 + 12i - 12i - 9i^2 \)
5Step 5: Simplify the products
Simplify using \( i^2 = -1 \):Numerator:\(24 + 18i + 8i + 6(-1) = 24 + 26i - 6 = 18 + 26i \) Denominator:\(16 + 12i - 12i - 9(-1) = 16 + 9 = 25 \)
6Step 6: Write the expression in the form \(a+bi\)
Divide each term in the numerator by the denominator to simplify:\( \frac{18 + 26i}{25} = \frac{18}{25} + \frac{26}{25}i \) Hence, the expression is \( \frac{18}{25} + \frac{26}{25}i \).
Key Concepts
ConjugateRationalizing the DenominatorImaginary Unit
Conjugate
Dealing with complex numbers often requires using the conjugate. The conjugate of a complex number is created by changing the sign of its imaginary part. For example, if you have a complex number like \( 4 - 3i \), its conjugate is \( 4 + 3i \).
Conjugates are essential in simplifying complex expressions, especially when we need to "rationalize" a denominator that is a complex number. By multiplying the numerator and denominator by the conjugate of the denominator, we effectively eliminate the imaginary unit \(i\) from the denominator. This transformation turns the denominator into a real number.
This is because when you multiply a complex number by its conjugate, the result is a real number. For instance, multiplying \( 4 - 3i \) and \( 4 + 3i \) gives \( 16 + 9 = 25 \) after simplification because \( i^2 \) equals \( -1 \). This is why using conjugates works so well in rationalizing denominators.
Conjugates are essential in simplifying complex expressions, especially when we need to "rationalize" a denominator that is a complex number. By multiplying the numerator and denominator by the conjugate of the denominator, we effectively eliminate the imaginary unit \(i\) from the denominator. This transformation turns the denominator into a real number.
This is because when you multiply a complex number by its conjugate, the result is a real number. For instance, multiplying \( 4 - 3i \) and \( 4 + 3i \) gives \( 16 + 9 = 25 \) after simplification because \( i^2 \) equals \( -1 \). This is why using conjugates works so well in rationalizing denominators.
Rationalizing the Denominator
Rationalizing the denominator is a technique used to eliminate imaginary numbers from the denominator of a fraction. This process simplifies the expression into a more standard form, making it easier to work with.
In the exercise, we have \( \frac{6+2i}{4-3i} \). Our goal is to rewrite it in the form \( a + bi \) where both \( a \) and \( b \) are real numbers. To achieve this, we multiply both the numerator and the denominator by the conjugate of the denominator, \( 4 + 3i \).
In the exercise, we have \( \frac{6+2i}{4-3i} \). Our goal is to rewrite it in the form \( a + bi \) where both \( a \) and \( b \) are real numbers. To achieve this, we multiply both the numerator and the denominator by the conjugate of the denominator, \( 4 + 3i \).
- Step 1: Multiply the numerator and denominator by the conjugate: \( (6+2i)(4+3i) \) and \( (4-3i)(4+3i) \).
- Step 2: When multiplied out, the denominator becomes \( 25 \) because \( 16 + 9 \) after simplifying \( -9i^2 \). The numerator is expanded to \( 18 + 26i \) by using the distributive property and substituting \( i^2 \) with \( -1 \).
- Step 3: Simplify the resulting expression \( \frac{18 + 26i}{25} \) to the form \( \frac{18}{25} + \frac{26}{25}i \).
Imaginary Unit
The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the real numbers. It is defined by the property \( i^2 = -1 \). This concept is essential for understanding complex numbers, which are numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
When dealing with expressions involving complex numbers, knowing how to manipulate \(i\) is crucial. For instance, in the original exercise, the term \( 6i^2 \) appears during expansion, which simplifies to \(-6\) because \( i^2 = -1 \). This substitution is pivotal in handling complex numbers.
When dealing with expressions involving complex numbers, knowing how to manipulate \(i\) is crucial. For instance, in the original exercise, the term \( 6i^2 \) appears during expansion, which simplifies to \(-6\) because \( i^2 = -1 \). This substitution is pivotal in handling complex numbers.
- Simplifying Expressions: Change \( i^2 \) to \(-1\) when it appears to reduce the complexity of calculations.
- Working with Powers of \(i\): It’s useful to remember that \( i^3 = -i \) and \( i^4 = 1 \), as these cycle through powers of \(i\).
Other exercises in this chapter
Problem 46
Solve. \(\sqrt{7 x-4}=\sqrt{4-7 x}\)
View solution Problem 46
Use the properties of exponents to simplify each expression. Write with positive exponents. See Example 4 . $$ 5^{1 / 2} \cdot 5^{1 / 6} $$
View solution Problem 46
Simplify. See Examples 3 and 4 $$ \sqrt[3]{y^{5}} $$
View solution Problem 46
Rationalize each denominator. See Example 4. $$ \frac{\sqrt{3}+\sqrt{4}}{\sqrt{2}-\sqrt{3}} $$
View solution