Problem 51 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-2} \sqrt{-27}$$

Step-by-Step Solution

Answer

\(\sqrt{-2} \sqrt{-27} = -3\sqrt{6}\).

Step 1: Express in terms of i

First, express each square root of a negative number using the imaginary unit, \(i\). Remember, \(\sqrt{-a} = i\sqrt{a}\).So, \(\sqrt{-2} = i\sqrt{2}\) and \(\sqrt{-27} = i\sqrt{27}\).

Step 2: Multiply the expressions

Now, multiply the expressions obtained in Step 1:\[(i\sqrt{2})(i\sqrt{27})\]This can be simplified using the property \(i^2 = -1\).

Step 3: Simplify the expression

Continue simplification by combining and simplifying:\[(i\sqrt{2})(i\sqrt{27}) = i^2\sqrt{2}\sqrt{27} = (-1)\sqrt{54}\]Now, focus on simplfying \(\sqrt{54}\).

Step 4: Simplify the square root

The number \(54\) can be factored as \(54 = 9 \times 6\), so:\[\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9}\sqrt{6} = 3\sqrt{6}\]Finally, apply this to the expression from Step 3.

Step 5: Final simplification

Substitute \(\sqrt{54} = 3\sqrt{6}\) back into the expression to get:\[(-1)\sqrt{54} = -1 \times 3\sqrt{6} = -3\sqrt{6}\]Thus, the expression \(\sqrt{-2} \sqrt{-27}\) simplifies to \(-3\sqrt{6}\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers. It is represented by the letter \(i\). The important characteristic of \(i\) is that \(i^2 = -1\). This property allows us to handle square roots of negative numbers. Without \(i\), we couldn't take square roots of negative values, as they don't have real-number solutions.

• For example, \(\sqrt{-1} = i\).
• Thus, any negative square root can be expressed using \(i\). For instance, \(\sqrt{-4} = i\sqrt{4} = 2i\).

This conversion is crucial in simplifying and performing operations with complex numbers, as it transforms otherwise problematic expressions into manageable forms. Through these conversions, we can apply regular algebraic rules to a wider range of problems.

Simplifying Radicals

When simplifying radicals, the goal is to express the radical in its simplest form. This involves dealing with the factors inside the square root and breaking them down where possible.
Start by factoring the number under the square root to find perfect squares. Perfect squares are numbers like 4, 9, 16, etc., whose square roots are whole numbers.

• For example, consider \(\sqrt{54}\). Factoring 54 gives \(54 = 9 \times 6\).
• Since \(9\) is a perfect square, \(\sqrt{9} = 3\).

This allows us to simplify \(\sqrt{54}\) as follows: \(\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9}\sqrt{6} = 3\sqrt{6}\).
By recognizing and extracting perfect squares from under the root, we achieve a simplified form that is easier to work with, especially in further calculations.

Multiplying Radicals

Multiplying radicals involves combining two or more radical expressions. The main rule to remember is that you can multiply the numbers inside the radicals, then if possible the resulting radical is simplified.
When multiplying, also consider the imaginary unit if it is involved. Here’s how you handle multiplying radicals with imaginary numbers:

• If you have \((i\sqrt{a})(i\sqrt{b})\), you first multiply the radicals: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).
• Then, deal with the \(i\) terms: \(i \times i = i^2\). Since \(i^2 = -1\), this becomes \(-1 \times \sqrt{ab}\).

This shows that the operation isn’t just about multiplying values but also re-arranging and simplifying to reflect real and imaginary contributions clearly. Practicing this multiplication helps in solving complex algebraic problems, and it ensures the expressions are in their simplest and most interpretable forms. For example, in our concrete case, \((i\sqrt{2})(i\sqrt{27}) = -\sqrt{54} = -3\sqrt{6}\).