Problem 63 - Precalculus Student Solutions Manual 5th

Question

Find each product. Write the answer in standard form. $$i(3-4 i)(3+4 i)$$

Step-by-Step Solution

Answer

The answer in standard form is 25i.

Step 1: Recognize the form

Notice that \(3 - 4i\) and \(3 + 4i\) are conjugates of each other. When multiplied together, they follow the pattern of \( (a - bi)(a + bi) \) which results in \( a^2 + b^2 \).

Step 2: Apply the pattern

Here, we have \( a = 3 \) and \( b = 4 \). Using the pattern, we get: \[ (3 - 4i)(3 + 4i) = 3^2 + 4^2 = 9 + 16 = 25 \]

Step 3: Multiply by i

Now multiply the result by \( i \): \[ i(3 - 4i)(3 + 4i) = i \times 25 = 25i \]

Step 4: Write in standard form

The standard form of a complex number is \( a + bi \). Our result is already in the form \( 0 + 25i \) which simplifies to \( 25i \).

Key Concepts

Conjugate

In complex numbers, the complex conjugate of a given number is obtained by changing the sign of the imaginary part. For instance, if you have a complex number written as \(a + bi\), its conjugate is \(a - bi\). Here, \(a\) is the real part, and \(b\) is the imaginary part.

Conjugates help simplify multiplication and division of complex numbers. Notice that when multiplying a complex number by its conjugate, the result is always a real number. For example, \( (3 - 4i) \times (3 + 4i) = 3^2 + 4^2 = 9 + 16 = 25 \).

This property results from the pattern \((a - bi)(a + bi) = a^2 + b^2\). Remember that the multiplication of \(-bi \) and \(+bi\) leads to \(-b^2i^2\). Since \(i^2 = -1\), you get \(-b^2(-1) = b^2\), eliminating the imaginary unit and leaving a real number.

Standard Form

The standard form of a complex number is \(a + bi\), where \((a) \) represents the real part and \((bi)\) represents the imaginary part. Ensuring complex numbers are in this form makes them easier to interpret and work with.

For the exercise given, we start with \(3 - 4i\) and \(3 + 4i\) and recognize them as conjugates. Using the multiplication result of \((3 - 4i)(3 + 4i) = 25\) from the steps, we get a real number.

Finally, multiplying this by \(i\), we derive \(25i\), represented in standard form as \(0 + 25i\). Here, \(0\) is the real part, and \((25i)\) is the imaginary part.

Imaginary Unit

The imaginary unit, denoted as \((i)\), is a mathematical concept to extend the real numbers system. Defined by \((i^2 = -1)\), it's essential in complex numbers.

When dealing with complex numbers, it's crucial to understand how \((i)\) behaves in operations. For example, in the exercise, after calculating \((3 - 4i)(3 + 4i) = 25\), multiplying by \((i)\) gives \'\(25i\)\ \, keeping the result aligned with standard form conventions.

Complex numbers often use \((i)\) in real-world applications like engineering and physics to solve problems involving wave functions, electrical circuits, and more. It’s also foundational for fields dealing with two-dimensional numbers and rotations.