Problem 33 - 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. $$ \frac{\sqrt{-36} \sqrt{-49}}{\sqrt{-16}} $$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(0 - 10.5i\).

Step 1: Simplify each square root

First, remember that \( \sqrt{-a} = \sqrt{a}i \). Using this property, simplify each square root in the expression: \( \sqrt{-36} = \sqrt{36}i = 6i \), \( \sqrt{-49} = \sqrt{49}i = 7i \), and \( \sqrt{-16} = \sqrt{16}i = 4i \).

Step 2: Multiply the simplified expressions in the numerator

Multiply \( 6i \) and \( 7i \) to get the numerator. \( 6i \times 7i = 42i^2 \). Since \( i^2 = -1 \), this simplifies to \( 42 (-1) = -42 \).

Step 3: Divide by the denominator

Now, divide the result from Step 2 by the denominator: \( \frac{-42}{4i} \). Simplify by dividing each part: \(-\frac{42}{4}i = -10.5i\).

Step 4: Convert to the form \(a + bi\)

Since there is no real number part, we express \(-0 - 10.5i\) as \(a + bi\), where \(a = 0\) and \(b = -10.5\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fundamental part of complex numbers, which are used to solve equations that simple real numbers cannot. An imaginary number is created when you take the square root of a negative number. Instead of resulting in a real number, it results in a number with an 'i,' where 'i' is the imaginary unit. For example, \( \sqrt{-1} = i \). Imaginary numbers are crucial because they allow us to extend the concept of numbers and tackle problems involving square roots of negatives. In the expression given, we transformed real numbers into imaginary numbers with operations like \( \sqrt{-36} = 6i \). Imaginary numbers help escape the limitations of traditional mathematics and open new realms of problem-solving.

Square Roots

Square roots are a mathematical operation that finds a number, which when multiplied by itself, gives the original number. For positive numbers, this is straightforward. However, when it comes to negative numbers, things change because a real number when squared never gives a negative result. This is why imaginary numbers are essential. They provide the tool to define square roots of negative numbers.

• For any negative number \(-a\), the square root is \( \sqrt{-a} = \sqrt{a}i \).
• This property, \( \sqrt{-1} = i \), is key in arranging complex numbers into a form easy to work with.
• Using this property, we simplified expressions like \( \sqrt{-36} \) into imaginary numbers in the original exercise.

Thus, converting square roots of negative numbers into imaginary numbers makes equations solvable that otherwise would not be.

Simplification of Expressions

Simplification is an essential step in solving mathematical problems to make them easier to interpret and work with. For complex numbers, simplification often means reducing expressions to the form \(a + bi\), where \(a\) and \(b\) are real numbers. This standard form is straightforward and immediately tells us about the structure and values involved.

Consider the original expression we simplified:

• We started by simplifying each square root using the imaginary unit \(i\).
• Then, we multiplied the outputs for the numerator: \(6i \times 7i = 42i^2 \), recalling that \(i^2 = -1\), which simplifies to \(-42\).
• Finally, we divided by the simplified denominator to get the expression in the standard form \(a + bi\).

The entire process of simplification transforms a complex expression into a more manageable form, making it easier to understand and solve.