Problem 50 - Precalculus Mathematics for Calculus

Question

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{\frac{1}{3}} \sqrt{-27}$$

Step-by-Step Solution

Answer

The result of the expression is \(0 + 3i\).

Step 1: Simplify the Radical Expression

Start by splitting the expression into two individual square roots: \( \sqrt{\frac{1}{3}} \) and \( \sqrt{-27} \). The first can be rewritten as \( \frac{1}{\sqrt{3}} \), and the latter can be simplified by removing the negative under the square root as \( \sqrt{27} \cdot i \).

Step 2: Simplify \( \sqrt{27} \)

The expression \( \sqrt{27} \) can be simplified by factoring 27 into 9 and 3, where 9 is a perfect square. Therefore, \( \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3} \).

Step 3: Combine the Simplified Parts

The expression becomes \( \frac{1}{\sqrt{3}} \cdot 3\sqrt{3} \cdot i \). Using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), simplify \( \frac{1}{\sqrt{3}} \cdot 3\sqrt{3} = \frac{3\sqrt{3} \cdot i}{\sqrt{3}} \).

Step 4: Simplify \( \frac{3\sqrt{3}}{\sqrt{3}} \)

Cancel \( \sqrt{3} \) in the numerator and denominator: \( \frac{3\sqrt{3}}{\sqrt{3}} = 3 \). Thus, the expression simplifies to \( 3i \).

Step 5: Express the Final Answer in the Form \( a+bi \)

The final result is expressed in the form \( a+bi \) as \( 0+3i \).

Key Concepts

Radical Expressions

Radical expressions may seem intimidating at first, but they are quite simple once you break them down. A radical expression involves roots, such as square roots or cube roots. In mathematics, the most common radical expression is the square root.
To simplify a radical expression like \(\sqrt{\frac{1}{3}}\), we can rewrite it as \(\frac{1}{\sqrt{3}}\). By practicing these conversions, dealing with radicals becomes more straightforward.

• When you encounter radical expressions, always look to simplify them. This often involves breaking numbers down into factors, especially perfect squares.
• Remember that it is possible to split a radical expression into smaller parts for easier simplification, just as we did by separating \(\sqrt{\frac{1}{3}}\) and \(\sqrt{-27}\).

Understanding and simplifying radical expressions aids in solving more complex problems and forms a foundational element in algebra and higher mathematics.

Imaginary Unit

The imaginary unit, represented by \(i\), is a crucial element when dealing with square roots of negative numbers. By definition, \(i = \sqrt{-1}\), allowing us to express roots of negative numbers in terms of \(i\). For instance, in the expression \(\sqrt{-27}\), we rewrite it as \(\sqrt{27} \cdot i\).
This process is essential in transforming otherwise non-real roots into a manageable form for further calculations.

• Imaginary numbers facilitate a consistent method for handling operations involving the square roots of negative numbers.
• It's important to remember that the square of \(i\) is \(-1\), forming the basis for many operations in complex numbers.

The imaginary unit helps expand our number system beyond the real numbers, allowing the graphical representation and calculation of complex numbers on the complex plane.

Simplifying Square Roots

Simplifying square roots is a vital skill in algebra that involves breaking down a number into components to extract a perfect square. Consider \(\sqrt{27}\), where 27 can be decomposed into \(9 \times 3\). Here, 9 is a perfect square, leading to the simplification \(3\sqrt{3}\).
The key steps are:

• Identify any perfect square factors of the number under the square root.
• Extract the square root of those perfect squares to simplify the expression.

This simplification process is crucial not only in basic algebra but also in more advanced mathematical computations involving radical expressions and complex numbers.