Problem 77 - Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i\) $$ 3-\sqrt{-64} $$

Step-by-Step Solution

Answer

The simplified form is \(3 - 8i\).

Step 1: Identify the Complex Part

Notice the term \(-\sqrt{-64}\). Since the square root of a negative number introduces an imaginary unit \(i\), rewrite \(\sqrt{-64}\) as \(\sqrt{64} \cdot \sqrt{-1}\).

Step 2: Simplify the Square Root

Calculate \(\sqrt{64}\), which is 8, hence \(-\sqrt{-64} = -8i\) because \(\sqrt{-1} = i\).

Step 3: Substitute Back into the Original Expression

Replace \(-\sqrt{-64}\) with \(-8i\) in the expression \(3 - \sqrt{-64}\). The expression becomes \(3 - 8i\).

Step 4: Write the Result in the Form \(a + bi\)

The expression \(3 - 8i\) is already in the desired form \(a + bi\), where \(a = 3\) and \(b = -8\).

Key Concepts

Imaginary Numbers

When dealing with complex numbers, the term 'imaginary numbers' often surfaces. Imaginary numbers arise when we take the square root of a negative number, which isn't possible within the realm of real numbers. This is where the 'imaginary unit' denoted by the symbol \(i\) comes into play. The imaginary unit is defined as:

• \(i = \sqrt{-1}\)
• This means \(i^2 = -1\)

So, when you encounter the square root of a negative number, you can rewrite it using \(i\). For example, for \(-\sqrt{-64}\), you treat it as \(-\sqrt{64} \cdot \sqrt{-1}\), which simplifies to \(-8i\). Understanding this fundamental concept allows you to work confidently with complex numbers.

Simplification

Simplification, particularly in the context of complex numbers, is about rewriting expressions to their simplest form. This often involves identifying and isolating different parts of an expression, such as real and imaginary components.

Here's how simplification works for the given problem:

• Start with the expression \(3 - \sqrt{-64}\).
• Recognize \(-\sqrt{-64}\) as the imaginary part and rewrite it using \(i\). Convert \(\sqrt{-64}\) to \(8i\) (identifying that \(\sqrt{64} = 8\) and \(\sqrt{-1} = i\)).
• This gives us \(-8i\).
• Substitute \(-8i\) back into the original expression, turning it into \(3 - 8i\).

This shows how identifying parts of a complex expression and rewriting them using core concepts can simplify the problem, making it easier to work with further.

Complex Conjugates

Although not directly used in solving the given problem, the concept of complex conjugates is vital to understanding complex numbers more deeply. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. If you have a complex number \(a + bi\), its conjugate is \(a - bi\).

Why are complex conjugates important?

• They are helpful in simplifying divisions involving complex numbers.
• Multiplying a complex number by its conjugate results in a real number (useful for rationalizing denominators).
• For example, multiplying \((a + bi)(a - bi) = a^2 + b^2\), which is a real number.

Communicating the concept of complex conjugates makes it clear how powerful they are in manipulating complex expressions and solving equations involving complex numbers.