Problem 44 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (-7+\sqrt{-81})-(-2-\sqrt{-64}) $$

Step-by-Step Solution

Answer

The answer is \(-5 + 17i\).

Step 1: Simplify the Square Roots

First, we need to simplify the square roots in the expression. The square root of a negative number is expressed in terms of the imaginary unit, \(i\), where \(i = \sqrt{-1}\). Hence, \(\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i\). Similarly, \(\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i\).

Step 2: Substitute Simplified Square Roots

Substitute the simplified square roots back into the original expression: \((-7+9i) - (-2-8i)\).

Step 3: Distribute the Negative Sign

Simplify the expression by distributing the negative sign in the subtracted expression: \((-7+9i) + (2+8i)\).

Step 4: Combine Real Parts

Combine the real parts of the complex numbers: \(-7 + 2 = -5\).

Step 5: Combine Imaginary Parts

Combine the imaginary parts of the complex numbers: \(9i + 8i = 17i\).

Step 6: Write Final Answer

Combine the results from Steps 4 and 5 to express the answer in standard form: \(-5 + 17i\).

Key Concepts

Understanding the Imaginary Unit

In mathematics, the imaginary unit is denoted by the symbol \(i\). It is defined as the square root of negative one, meaning that \(i = \sqrt{-1}\). This concept is fundamental when working with complex numbers, as it allows us to solve equations that don't have real solutions. For example, the equation \(x^2 + 1 = 0\) has no real solution because there is no real number that satisfies \(x^2 = -1\). However, by introducing the imaginary unit \(i\), we can express the solutions as \(x = i\) and \(x = -i\). When dealing with negative square roots, remember:

• The square root of a negative number \(\sqrt{-a}\) is expressed as \(\sqrt{a} \cdot i\).
• \(i^2 = -1\), which is a key property of the imaginary unit.

By understanding this building block, you can start solving more complex expressions that involve imaginary numbers.

Simplifying Square Roots of Negative Numbers

Square roots of negative numbers can seem perplexing at first, but with the help of the imaginary unit \(i\), they become manageable. In essence, the square root of a negative number is transformed using \(i\). For instance:- To simplify \(\sqrt{-81}\), break it down into components:

• Identify \(\sqrt{81} = 9\).
• Multiply by \(\sqrt{-1} = i\).
• Combine them to get \(\sqrt{-81} = 9i\).

Similarly, for \(\sqrt{-64}\), the process is:

• Recognize \(\sqrt{64} = 8\).
• Use \(\sqrt{-1} = i\).
• Combine these to find \(\sqrt{-64} = 8i\).

This method simplifies working with complex square roots and is crucial in performing complex number operations.

Operations with Complex Numbers

Complex number operations include addition, subtraction, and multiplication, allowing you to combine and manipulate expressions involving complex numbers. A complex number is generally expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.When subtracting complex numbers like the example \((-7+\sqrt{-81})-(-2-\sqrt{-64})\), follow these steps:

• First, substitute the simplified square roots: \((-7 + 9i) - (-2 - 8i)\).
• Distribute the negative sign: \((-7 + 9i) + (2 + 8i)\).
• Add the real parts together: \(-7 + 2 = -5\).
• Combine the imaginary parts: \(9i + 8i = 17i\).
• The final expression is \(-5 + 17i\), which is the result of the operation.

Mastering these operations allows for deeper understanding and manipulation of complex numbers in mathematical problems.