Problem 81 - Intermediate Algebra

Question

Divide. Write all answers in the form \(a+b i.\) $$ \frac{8+\sqrt{-144}}{2+\sqrt{-9}} $$

Step-by-Step Solution

Answer

The answer is \(4 + 0i\).

Step 1: Simplify the Imaginary Numbers

Start by simplifying the square roots of negative numbers using the imaginary unit, where \sqrt{-1} = i\. \( \sqrt{-144} = i \cdot \sqrt{144} = 12i \) and \( \sqrt{-9} = i \cdot \sqrt{9} = 3i \). Substitute these into the original problem to get: \( \frac{8 + 12i}{2 + 3i} \).

Step 2: Multiply by the Conjugate

To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2 + 3i \) is \( 2 - 3i \). Multiply: \( \frac{(8 + 12i)(2 - 3i)}{(2 + 3i)(2 - 3i)} \).

Step 3: Simplify the Denominator

The denominator is a difference of squares: \( (2 + 3i)(2 - 3i) = 4 - (3i)^2 = 4 - 9(-1) = 4 + 9 = 13 \).

Step 4: Expand the Numerator

Use the distributive property (FOIL method) to expand the numerator: \((8 + 12i)(2 - 3i) = 8 \cdot 2 + 8(-3i) + 12i(2) - 12i(3i) = 16 - 24i + 24i - 36i^2\). Simplify terms: \(-24i + 24i = 0 \), and \(-36i^2 = 36 \). Thus, the numerator becomes \(16 + 36 = 52\).

Step 5: Divide to Find the Result

Divide the real part of the expanded result by the denominator: \( \frac{52}{13} = 4 \). The imaginary part is zero since \(-24i + 24i = 0\), so the result is \(4 + 0i\).

Step 6: Finalize the Answer

Combine the real and imaginary components to finalize the result. Since the imaginary component is zero, the simplified answer is: \(4 + 0i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a key part of complex numbers, which are numbers that have both a real and an imaginary component. An imaginary number is formed when taking the square root of a negative number. This occurs frequently within mathematics and various applications. The imaginary unit, denoted by \(i\), is defined as \(i = \sqrt{-1}\). When we have a negative square root, such as \(\sqrt{-144}\), we convert it to an imaginary number by recognizing it as \(12i\), since \(\sqrt{144} = 12\) multiplied by \(i\).
Similarly, \(\sqrt{-9}\) can be expressed as \(3i\). Combining these concepts allows complex numbers to be simplified into the standard form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Conjugate

When working with complex numbers, the conjugate plays an important role, especially when dividing complex numbers. For any complex number \(a + bi\), its conjugate is \(a - bi\). Using conjugates is crucial for removing imaginary parts from the denominators of fractions involving complex numbers.
For instance, if we need to divide by \(2 + 3i\), we use its conjugate, \(2 - 3i\), because multiplying these two results in a real number. This simplifies the division and makes the calculations easier. Multiplying by the conjugate effectively forces the imaginary part to zero in the denominator by relying on the algebraic identity of the difference of squares.

FOIL Method

The FOIL method is an algebraic tool used to simplify the multiplication of two binomials. FOIL stands for First, Outer, Inner, Last, and it refers to the order of operations in multiplying each term in the binomials. This method is very useful when dealing with complex numbers.
Let's break it down using these binomials: \((8 + 12i)\) and \((2 - 3i)\). By FOILing, we first multiply the First terms: \(8 \times 2 = 16\). Then the Outer terms: \(8 \times -3i = -24i\). Next, the Inner terms: \(12i \times 2 = 24i\). And finally, the Last terms: \(12i \times -3i = -36i^2\).
Combining these results, and remembering that \(i^2 = -1\), transforms \(-36i^2\) into \(36\), giving a simplified result of \(16 + 36 = 52\) after combining like terms.

Difference of Squares

The difference of squares is a mathematical identity that simplifies the multiplication of two binomial expressions. It states that \((a + b)(a - b) = a^2 - b^2\). This is especially beneficial when simplifying the division of complex numbers by making the denominator real.
When applied to the division of complex numbers, multiplying by the conjugate makes use of this identity: for \((2 + 3i)(2 - 3i)\), substituting into the identity gives us \[ (2)^2 - (3i)^2 = 4 - 9i^2 \].
Since \(i^2 = -1\), the expression simplifies to \(4 + 9 = 13\). This operation efficiently turns the denominator into a real number, facilitating the division process without any remaining imaginary parts.