Problem 37 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{\frac{11}{9}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{11}}{3} \)

Step 1: Split the Radical

Recognize that the radical expression involves a fraction. According to the property of radicals, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), split the given radical: \( \sqrt{\frac{11}{9}} = \frac{\sqrt{11}}{\sqrt{9}} \).

Step 2: Simplify the Denominator

Simplify the square root in the denominator of the fraction. Compute \( \sqrt{9} \), which equals 3, so the expression becomes \( \frac{\sqrt{11}}{3} \).

Step 3: Simplify the Expression Further

The expression \( \frac{\sqrt{11}}{3} \) is already in its simplest form because \( \sqrt{11} \) cannot be simplified further as 11 is a prime number. Therefore, the final answer is simply \( \frac{\sqrt{11}}{3} \).

Key Concepts

Simplifying Radicals

Simplifying radicals is all about making complex expressions easier to handle. When we talk about radicals, think of the square root symbol, known as the radical sign. It works hand-in-hand with indices (numbers or variables under the radical sign) that need to be simplified. The key to simplification is understanding that we can break down expressions into smaller parts, making them easier to work with.

To simplify involves breaking down complicated radical expressions. Often, this means making the numbers inside the radical as small as possible while still having the same value. For instance, with square roots, you look for perfect squares inside the number. Doing so means finding the largest perfect square factor and writing the expression as the product of two square roots.

• For example, to simplify a number like \( \sqrt{50} \), you recognize that 50 is \( 25 \times 2 \), hence \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \).
• In the case of fractions, use the property of radicals \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This step is often essential in simplifying radical expressions involving fractions.

By consistently applying these techniques, you can turn any radical expression into its simplest and most manageable form.

Properties of Radicals

Understanding the properties of radicals provides the foundation for working with these expressions. These properties extend from basic arithmetic laws applied to roots and remain consistent regardless of the numbers involved.

A crucial principle is the product property of square roots \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This property allows you to break down radicals into smaller parts and is especially handy when simplifying. The quotient property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) is equally important, providing a method for handling radicals involving fractions.

• The quotient property applies when both the numerator and the denominator are under a single radical sign.
• These properties make it easier to work around expressions and seek their simplest form.

Be mindful that operations like addition and subtraction of radicals require identical radicands (values under the radical), akin to combining like terms in algebra. This consistency ensures clarity in manipulation and simplification.

Square Roots

Square roots are the most common type of radical. They represent a value that, when multiplied by itself, gives the original number. The square root of a number \( n \) is expressed as \( \sqrt{n} \). Understanding how square roots work is pivotal for radical simplification.

A few special cases are handy, such as perfect squares, which are numbers that are squares of integers (like 1, 4, 9, 16, 25). Recognizing these can instantly simplify calculations.

• For instance, \( \sqrt{9} = 3 \) because 9 is 3 times 3.
• When you identify the perfect squares, simplify them directly without further calculation.

The nature of square roots makes them challenging at times, especially with non-perfect squares. Unlike perfect squares, numbers that aren’t perfect squares don’t simplify to an integer, and you’re often left with an irrational number. For example, \( \sqrt{11} \) cannot be simplified further since 11 isn’t a perfect square or factorable into smaller perfect squares.

Understanding that some expressions will not simplify further, like in our original problem \( \sqrt{11} \), is just as valuable as knowing how to do the calculations.