Problem 40 - Precalculus Student Solutions Manual 5th

Question

Write each number in standard form a \(+b i\) $$\frac{20+\sqrt{-8}}{2}$$

Step-by-Step Solution

Answer

10 + \( \sqrt{2}i \)

Step 1: Simplify the expression under the square root

Identify the square root of the negative number \( \sqrt{-8} \). Note that \( \sqrt{-1} = i \) where \( i \) is the imaginary unit. So, \( \sqrt{-8} = \sqrt{8 \cdot (-1)} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2} \cdot i \).

Step 2: Substitute the simplified square root into the expression

Replace \( \sqrt{-8} \) in the original expression with \( 2\sqrt{2} i \). The expression becomes \( \frac{20 + 2\sqrt{2}i}{2} \).

Step 3: Divide each term by the denominator

Divide both the real part and the imaginary part by 2. This yields \( \frac{20}{2} + \frac{2\sqrt{2}i}{2} = 10 + \sqrt{2}i \).

Key Concepts

imaginary unit

In complex numbers, the imaginary unit is denoted by the letter 'i'. It is defined by the fundamental property that \( i^2 = -1 \). This property allows us to extend the real number system into the complex number system.

Imaginary units are particularly useful when dealing with square roots of negative numbers. Normally, the square root of a negative number doesn't exist within the set of real numbers. However, by introducing the imaginary unit 'i', we can simplify such expressions. For example, \( \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2} \cdot i \). This way, we can handle and simplify complex expressions.

Remember, the imaginary unit follows its own rules when it comes to arithmetic. For example:

• i * i = -1
• i * (-i) = 1
• i^3 = -i
• i^4 = 1

simplifying square roots

Simplifying square roots involves expressing a number inside the radical sign in a more manageable form. When the number under the square root is negative, you also have to use the imaginary unit 'i'.

Let's step through how to simplify \( \sqrt{-8} \):

- First, express the negative number under the square root as a product: \( -8 = 8 \cdot -1 \).- Next, split the square root into two separate roots: \( \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} \).- With \( \sqrt{-1} \) defined as 'i', this becomes: \( 2\sqrt{2} \cdot i \).
This simplification helps incorporate the imaginary unit into our expression.
To further simplify square roots:

• Factor the number into primes.
• Pair the primes if possible.
• Take one number out of the square root for each pair.
• Always include 'i' if there is a negative sign under the root.

Applying these steps rigorously can turn complex-looking expressions into simpler, manageable ones.

standard form of complex numbers

The standard form of a complex number is written as \ (a + bi) \. Here, 'a' represents the real part, and 'b' represents the coefficient of the imaginary part, with 'i' being the imaginary unit.

Take the example \( \frac{20 + \sqrt{-8}}{2} \). We rearrange this into standard form by simplifying it step by step.

First, simplify the square root:

• \( \sqrt{-8} = 2\sqrt{2} \cdot i \)

Next, substitute back into the original expression:

• \( \frac{20 + 2\sqrt{2}i}{2} \)

Finally, divide each term by 2 to get the standard form:

• \( 10 + \sqrt{2}i \)

Now the expression is in the form \( a + bi \), making it easier to work with.

Using the standard form is beneficial because it distinctly separates the real and imaginary parts. This separation simplifies addition, subtraction, and other operations with complex numbers.

To convert any complex expression into standard form:

• Simplify all square roots.
• Perform all necessary algebraic operations.
• Separate the real part from the imaginary part.