Problem 55 - Precalculus Mathematics for Calculus

Question

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{\sqrt{-36}}{\sqrt{-2} \sqrt{-9}}$$

Step-by-Step Solution

Answer

The expression evaluates to \(-i\sqrt{2}\).

Step 1: Evaluate the Square Roots

Start by evaluating the square roots of the negative numbers. Recall that \( \sqrt{-x} = i \sqrt{x} \). So, \( \sqrt{-36} = i \sqrt{36} = 6i \) and \( \sqrt{-9} = i \sqrt{9} = 3i \).

Step 2: Simplify the Denominators

For the expression \( \sqrt{-2} \times \sqrt{-9} \), apply the property \( \sqrt{a}\sqrt{b} = \sqrt{ab} \) if both terms are non-negative. Since both terms are negative, use \( \sqrt{-a} \times \sqrt{-b} = i\sqrt{a} \times i\sqrt{b} = (i^2)\sqrt{ab} = -\sqrt{ab} \). Thus, \( \sqrt{-2} \times \sqrt{-9} = -(\sqrt{18}) = -3\sqrt{2} \) because \( \sqrt{18} = 3\sqrt{2} \).

Step 3: Form the Division of Complex Numbers

Now substitute the simplified expressions into the original division. You have: \( \frac{6i}{-3\sqrt{2}} \). Simplify by dividing both the numerator and the denominator by \(3\), leading to \( \frac{2i}{-\sqrt{2}} \).

Step 4: Rationalize the Denominator

To express in standard form \(a + bi\), multiply both the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator: \( \frac{2i \times \sqrt{2}}{-2} = \frac{-2i\sqrt{2}}{2} = -i\sqrt{2} \).

Step 5: Express in Complex Form

The simplified expression \(-i\sqrt{2}\) is already in the form \(a + bi\). It implies \( a = 0 \) and \( b = -\sqrt{2} \).

Key Concepts

Complex Numbers

Complex numbers are fascinating and extend our number system beyond the familiar real numbers. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\).

They combine two components: a real part (\

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate radicals from the denominator of a fraction. This process helps simplify expressions and makes them easier to manipulate. When a denominator includes an irrational number or imaginary numbers as radicals, we multiply both the numerator and the denominator of the fraction by a suitable expression that will result in a rational number in the denominator.

In the context of complex numbers, rationalizing can involve multiplying by the conjugate or using the special property of imaginary numbers, as demonstrated in the original problem.

• Example: When given a fraction like \(\frac{2i}{-\sqrt{2}}\), we can rationalize by multiplying the numerator and denominator by \(\sqrt{2}\), getting \(\frac{2i\times\sqrt{2}}{-2}\).
• This simplifies to \(-i\sqrt{2}\), which is a much cleaner expression.

Rationalizing makes it easier to add, subtract, multiply, or divide complex fractions.

Imaginary Numbers

Imaginary numbers arise when we take the square root of a negative number. The fundamental imaginary unit is \(i\), with the defining equation \(i^2 = -1\). This leads to an imaginary number being any multiple of \(i\).

For instance, in the expression \(\sqrt{-36}\), we apply the concept of the imaginary unit: \(\sqrt{-36} = i\sqrt{36} = 6i\). This is because \(36\) is a positive number and its square root is 6. By introducing the imaginary unit, this operation becomes valid.

• With \(\sqrt{-9} = i\sqrt{9} = 3i\), you see another example where the imaginary unit converts an operation that isn't possible with only real numbers into one that makes sense.
• Understanding this is vital when working with complex numbers since imaginary numbers are the backbone of complex number calculations.

Imaginary numbers open a new dimension in mathematics, enabling solutions to equations that lack real number solutions.