Problem 58 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-72}}{\sqrt{-6}}$$

Step-by-Step Solution

Answer

The simplified expression is \(2\sqrt{3}\).

Step 1: Express Square Roots in Terms of i

First, rewrite the square roots of negative numbers in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\).\\[\sqrt{-72} = i \sqrt{72}\]\\[\sqrt{-6} = i \sqrt{6}\]

Step 2: Rewrite the Fraction Using Expressions with i

Write the original expression as a fraction using the expressions from Step 1.\[\frac{\sqrt{-72}}{\sqrt{-6}} = \frac{i \sqrt{72}}{i \sqrt{6}}\]

Step 3: Simplify the Fraction

Since the \(i\) terms in the numerator and the denominator cancel each other out, simplify the expression to:\[\frac{\sqrt{72}}{\sqrt{6}}\]

Step 4: Simplify the Radical Expression

Simplify the remaining expression by dividing the square roots:\[\frac{\sqrt{72}}{\sqrt{6}} = \sqrt{\frac{72}{6}} = \sqrt{12}\]

Step 5: Express \(\sqrt{12}\) in Simplest Radical Form

Write \(\sqrt{12}\) in its simplest radical form by factoring:\[\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\]

Key Concepts

Complex Numbers

Complex numbers might sound intimidating, but they're actually very helpful in solving problems involving square roots of negative numbers. When you hear "complex number," think of a number system that includes both real numbers and an imaginary part. The standard form is usually written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Here, \(i\) is known as the imaginary unit and is defined by the property \(i^2 = -1\).

This clever invention helps when you're working with square roots of negative numbers. For instance, the square root of \(-1\) can be expressed as \(i\), thus transforming any negative square root into a complex number that incorporates \(i\).

Understanding complex numbers opens new pathways in mathematics, allowing you to plot numbers on a plane, solve polynomial equations that have no "real" solutions, and perform operations with expressions containing negative roots. It's all about extending our understanding of numbers to a wider world.

Radical Expressions

Radical expressions involve roots of numbers, often represented with the radical sign \(\sqrt{}\), such as square roots or cube roots. The radical symbol indicates that you're looking for a number which, when multiplied by itself a certain number of times, gives you the original value under the radical.

When dealing with negative numbers under the radical, especially in exercises like the one above, you need to introduce the imaginary number \(i\). For example, \(\sqrt{-72}\) is rewritten as \(i\sqrt{72}\), using the property that \(\sqrt{-1} = i\).

It's useful to understand how to manipulate these expressions because it allows you to simplify equations and expressions in mathematics, making them more manageable. This can involve separating the imaginary part from the rest of the radical expression, helping clarify the solution.

Simplifying Fractions

When it comes to simplifying fractions in mathematics, the process involves reducing a fraction to its simplest form. This makes it easier to work with or interpret results. For the specific case in complex numbers and radical expressions, once you transform negative roots into expressions with \(i\), you can start simplifying.

In the given exercise, you simplify a fraction with radicals and cancel out common terms. This often includes managing factors of \(i\) first, as seen in the step where \(i\) cancels out in both numerator and denominator. Then, you're left with just simplifying the radical part.

Once the radicals are separated from the imaginary unit, you can simplify the root itself by factoring it down. For instance, \(\sqrt{72}/\sqrt{6}\) simplifies to \(\sqrt{12}\), further reduced to its simplest form \(2\sqrt{3}\) by factoring into perfect squares. Mastering these steps in fraction simplification helps in handling complex maths problems with ease.