Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: 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 simplifies to \(6 + 58i\).

Step 1: Simplify the Square Roots

First, identify which terms involve square roots of negative numbers: \( \sqrt{-25} \) and \( \sqrt{-36} \). Use the fact that \( \sqrt{-1}=i \), so \( \sqrt{-25} = 5i \) and \( \sqrt{-36} = 6i \). The expression becomes \( (-3 + 5i)(8 - 6i) \).

Step 2: Apply the Distributive Property

Multiply each term in \( (-3 + 5i) \) by each term in \( (8 - 6i) \). This involves four multiplications:1. \(-3 \times 8 = -24\)2. \(-3 \times (-6i) = 18i\)3. \(5i \times 8 = 40i\)4. \(5i \times (-6i) = -30i^2\)

Step 3: Simplify \(i^2\)

Recall that \(i^2 = -1\). So, \(-30i^2 = -30(-1) = 30\).

Step 4: Combine Real and Imaginary Parts

Combine all the real and imaginary parts:- The real parts: \(-24 + 30 = 6\)- The imaginary parts: \(18i + 40i = 58i\)Therefore, the expression becomes \(6 + 58i\).

Key Concepts

Imaginary Numbers

Imaginary numbers come into play when we have square roots of negative numbers. Normally, the square root of a negative number isn't defined in the system of real numbers. However, with the introduction of imaginary numbers, we can represent such roots using the imaginary unit, denoted as \(i\).
The core idea is: \(i\) is defined by the equation \(i^2 = -1\).
For any negative number, say \(-n\), its square root can be expressed as \(\sqrt{-n} = \sqrt{n} \cdot i\).
For example, in our problem, \(\sqrt{-25}\) is rewritten as \(5i\), and \(\sqrt{-36}\) becomes \(6i\).

• The imaginary unit \(i\) helps us perform calculations that involve negative square roots without getting stuck.
• When included in an expression, like \(a+bi\), \(b\) represents the imaginary part.

This imaginary aspect of numbers adds a whole new dimension to mathematics, allowing us to solve equations that previously seemed unsolvable.

Square Roots

Square roots involve finding a number which, when multiplied by itself, gives the original number.
For positive numbers, the square root is straightforward. For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).
For negative numbers, square roots are expressed using imaginary numbers.
This is because no real number squared will result in a negative number.
For instance, \(\sqrt{-25}\) turns into \(5i\) as described above.

• To simplify an expression with a square root of a negative number, remember to extract \(i\).
• Rewrite the term as a product of a real number and \(i\), such as \(\sqrt{-25} = 5i\).

Understanding square roots and how they transition into imaginary numbers is crucial for solving complex mathematical problems.

Distributive Property

The distributive property is a fundamental principle in algebra that specifies how multiplication is distributed over addition.
It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true.
This property is especially useful when dealing with expressions that involve multiple terms inside brackets.

• In our problem, the distributive property helps expand \((-3 + 5i)(8 - 6i)\).
• The expression turns into several products: \(-3 \times 8\), \(-3 \times -6i\), \(5i \times 8\), and \(5i \times -6i\).
• This makes it easier to combine and simplify terms to get the expression in the desired form \(a+bi\).

Overall, applying the distributive property effectively simplifies the multiplication process in algebraic expressions.

Real Numbers

Real numbers include all the numbers on the number line—both positive and negative, including zero.
They can be whole numbers, fractions, or decimals.
The real numbers are used to express magnitudes and measurements in real-life scenarios.

• In the context of our expression, \(-3\) and \(8\) are real numbers.
• When we combine terms, the real portions of the calculations lead us to the real part of the final solution.
• The result \(6 + 58i\) clearly shows separation into real (\(6\)) and imaginary \((58i)\) parts.

Real numbers form the backbone of arithmetic and are crucial for understanding more complex systems in mathematics.