Problem 40 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{22}{9}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{22}}{3} \) is the simplest radical form.

Step 1: Separate the Fraction

Begin by separating the square root of a fraction into the square root of the numerator divided by the square root of the denominator. This gives us \( \frac{\sqrt{22}}{\sqrt{9}} \).

Step 2: Simplify the Denominator

Simplify the denominator by taking the square root of the number. Since \( \sqrt{9} = 3 \), the expression becomes \( \frac{\sqrt{22}}{3} \).

Step 3: Check the Numerator

Check if the numerator \( \sqrt{22} \) can be simplified further. The number 22 can be factored into 2 and 11, both of which are prime, so \( \sqrt{22} \) is already in its simplest form.

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots, most commonly square roots. They help in expressing quantities in a more manageable form, especially when dealing with powers and roots that aren't whole numbers. At the core, a radical expression consists of a radical sign, the number or expression inside, referred to as the radicand, and sometimes an index, which shows the degree of the root. For square roots, the index is typically not written, as it is implicitly understood to be 2.

Simplifying radicals often involves breaking down the radicand into factors to see if any of these are perfect squares, allowing us to take them outside the radical sign. For instance,

• \(\sqrt{16} = 4\) because 16 is a perfect square, as 4 times 4 equals 16.

These simplifications are important because they lead to cleaner, easier-to-work-with expressions.

The Concept of Square Roots

A square root is a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

Square roots allow mathematicians and scientists to reverse the process of squaring a number, providing a tool to explore properties of numbers and solve equations involving squares.

• Square roots can be either positive or negative, but the principal square root is always taken as the non-negative number.
• In the exercise \(\sqrt{\frac{22}{9}}\), we separated the fraction to apply the square root to both the numerator and the denominator individually.

By knowing how to deal with square roots, you gain a skill that is applicable in many areas of math, including geometry and algebra.

Simplifying Radicals

Simplifying radicals means rewriting them in their simplest form. In our exercise, we had to simplify \(\sqrt{\frac{22}{9}}\) to its simplest radical form. Here’s a general guide on how you can simplify radicals:

• Separate the expression into the square root of the numerator and denominator if it's a fraction.
• Simplify each part individually; for example, in our denominator, we had \(\sqrt{9}\), which simplifies easily to 3.
• For the numerator \(\sqrt{22}\), check if the number can be factored into perfect squares. If not, like in this case, it's already in its simplest form.

The main goal of simplifying radicals is to make them easier to work with, so they are more useful in further operations. By practicing these steps, you can tackle any problem involving radicals with confidence.