Problem 30 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(-3+\sqrt{-25})(8-\sqrt{-36})$$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(6 + 58i\).

Step 1: Simplify the square roots of negative numbers

First, note that \( \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \), as \( i \) is the imaginary unit, so \( \sqrt{-1} = i \). Similarly, \( \sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i \).

Step 2: Substitute and expand the expression

Substitute the simplified form into the original expression:\((-3+5i)(8-6i).\)Now distribute (FOIL) the terms:\((-3 \times 8) + (-3 \times -6i) + (5i \times 8) + (5i \times -6i).\)

Step 3: Perform the multiplication

Begin by carrying out the multiplications: \((-3 \times 8) = -24,\)\((-3 \times -6i) = 18i,\)\((5i \times 8) = 40i,\)\((5i \times -6i) = -30i^2.\)

Step 4: Combine the terms

Add the results of the multiplications together:\(-24 + 18i + 40i - 30i^2.\)Combine the imaginary terms:\(-24 + 58i - 30i^2.\)

Step 5: Simplify using the property of imaginary unit

Recall that \( i^2 = -1 \). Substitute this into the expression:\(-24 + 58i - 30(-1).\)This simplifies to:\(-24 + 58i + 30.\)

Step 6: Finish the simplification

Add the real parts together:\(-24 + 30 = 6.\)Thus, the expression becomes\(6 + 58i.\)

Key Concepts

Imaginary Unit

In mathematics, encountering the square root of a negative number may seem perplexing, because no real number squared results in a negative number. To address this, mathematicians have introduced the concept of the imaginary unit, represented by the symbol \(i\). The primary property of \(i\) is that \(i^2 = -1\), creating a foundation to express and simplify complex numbers involving square roots of negative values.

A practical example involves simplifying square roots of negative numbers. For instance, \( \sqrt{-25} \) can be rewritten by recognizing it as \( \sqrt{25} \cdot \sqrt{-1} \). Knowing \( \sqrt{25} \) is 5 and \( \sqrt{-1} = i \), we conclude that \( \sqrt{-25} = 5i \). This interpretation allows any square root of a negative number to be expressed as a product of a real number and \(i\).

Understanding the imaginary unit is crucial for handling complex numbers and expressions that involve them. It enables the transition from ambiguous roots to comprehensible mathematical operations with an additional axis, beyond the usual real number line.

FOIL Method

The FOIL method stands for First, Outside, Inside, Last. It is a technique often used for multiplying two binomials, simplifying the multiplication process by ensuring that each term in the first binomial is multiplied by each term in the second binomial.

• First: Multiply the first terms of each binomial.\((-3) \times 8 = -24\)
• Outside: Multiply the outer terms.\((-3) \times (-6i) = 18i\)
• Inside: Multiply the inner terms.\((5i) \times 8 = 40i\)
• Last: Multiply the last terms.\((5i) \times (-6i) = -30i^2\)

Applying the FOIL method ensures all interaction between the terms is considered, producing a complete expanded expression that can be further simplified. This strategy is especially beneficial when dealing with complex numbers, since it allows systematic incorporation of both real and imaginary components.

Simplifying Expressions

Simplifying expressions, particularly with complex numbers, involves more than just performing arithmetic operations. It requires understanding properties such as that of the imaginary unit \(i\).

The expression \(-24 + 18i + 40i - 30i^2\) showcases the culmination of several operations. Here, simplification involves two key processes:

• Combine like terms: Add \(18i\) and \(40i\) to obtain \(58i\).
• Simplify using \(i^2 = -1\): Replace \(-30i^2\) with \(30\), because \(-30 \cdot (-1) = 30\).

Continuing from this simplification, the real parts \(-24\) and \(30\) combine to form \(6\). Hence, the final expression stands as \(6 + 58i\).

Approaching simplifications with a clear understanding of the underlying principles ensures accuracy, simplifying complex numbers into straightforward forms, such as \(a + bi\), where \(a\) and \(b\) are real numbers.