Problem 45 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \sqrt{-3} \sqrt{-12} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-6 + 0i\).

Step 1: Express in terms of imaginary unit

Recognize that the square roots involve negative numbers and rewrite each square root in terms of the imaginary unit. For \( \sqrt{-3} \), this equals \( \sqrt{3} i \). Similarly, \( \sqrt{-12} \) becomes \( \sqrt{12} i \).

Step 2: Apply the product rule for square roots

Using the property \( \sqrt{a} \sqrt{b} = \sqrt{a \times b} \), apply this rule to \( \sqrt{3} i \times \sqrt{12} i \) to get \( \sqrt{36} i^2 \).

Step 3: Simplify the square root

Compute \( \sqrt{36} \), which is \( 6 \). Thus the expression becomes \( 6 i^2 \).

Step 4: Evaluate \(i^2\) and simplify

Recall that \( i^2 = -1 \). Substitute \( i^2 \) with \(-1\) in the expression, giving us \( 6 \times -1 = -6 \).

Step 5: Write in standard form

Express the result in the form \( a + bi \), which is \( -6 + 0i \).

Key Concepts

Imaginary Unit

Imaginary numbers are one of the fundamental concepts of complex numbers. These numbers are expressed using the imaginary unit, denoted by the symbol \(i\). Defined by the property \(i = \sqrt{-1}\), the imaginary unit allows us to take square roots of negative numbers, something that isn't possible with real numbers alone.
When working with square roots of negative numbers like \(\sqrt{-3}\) and \(\sqrt{-12}\), we utilize \(i\) to rewrite these expressions, resulting in \(\sqrt{3}i\) and \(\sqrt{12}i\) respectively.

• This step is crucial because it helps convert a potentially confusing expression into a workable form.
• Allows us to perform operations using the well-established rules of algebra, combined with the properties of \(i\).

Thus, understanding the imaginary unit is key to exploring and solving problems that involve complex numbers.

Product Rule for Square Roots

The product rule for square roots states that for any non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). However, things get a bit trickier when involving complex numbers since we're dealing with imaginary roots.
In problems like \(\sqrt{3}i \cdot \sqrt{12}i\), it’s important to carefully apply this rule:

• The expression transforms to \(\sqrt{3 \times 12} (i \cdot i)\).
• We get \(\sqrt{36} i^2\), considering the multiplication of the imaginary parts.

This process simplifies the computation and brings us one step closer to reaching the expression’s standard form. One must keep in mind the properties of \(i\), which include \(i^2 = -1\), making it integral in solving these problems.

Simplification

Simplification plays a major role in finding an understandable and neat form of the result that will be easily recognized and usable. After applying the product and understanding \(i\)'s properties, you often end up with an expression involving higher powers of \(i\) or simpler arithmetic operations.
In this exercise, after obtaining \(\sqrt{36} i^2\), calculate this further:

• \(\sqrt{36}\) simplifies to \(6\).
• Since \(i^2 = -1\), multiplying \(6\) by \(-1\) yields \(-6\).

Finally, expressing this result in standard form \(a+bi\), you write it as \(-6 + 0i\).
This approach not only simplifies complex calculations but also aligns the outcome clearly with the expected result format. This outcome aids in comprehension and further application within other mathematical concepts or real-world situations.