Problem 37 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( \frac{\sqrt{11}}{3} \).

Step 1: Understand the Radical Expression

The given expression is \( \sqrt{\frac{11}{9}} \). This is a square root of a fraction, where the numerator is 11 and the denominator is 9.

Step 2: Apply the Quotient Property of Square Roots

According to the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), we can separate the square root of a fraction into the quotient of the square roots of the numerator and the denominator. This means \( \sqrt{\frac{11}{9}} = \frac{\sqrt{11}}{\sqrt{9}} \).

Step 3: Simplify the Denominator

Find the square root of the denominator: \( \sqrt{9} = 3 \). So, the expression becomes \( \frac{\sqrt{11}}{3} \).

Step 4: Final Step: Confirmation

We see that \( \sqrt{11} \) cannot be simplified further as 11 is a prime number. Therefore, the expression \( \frac{\sqrt{11}}{3} \) is in its simplest form.

Key Concepts

Quotient Property of Square Roots

The Quotient Property of Square Roots is a valuable tool used to simplify expressions that involve the square root of a fraction. This property states that the square root of a fraction can be divided into the square roots of its numerator and its denominator separately. Suppose you have a fraction \( \frac{a}{b} \). The property can be mathematically expressed as:

• \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

By using this rule, the problem of taking the square root of a fraction becomes easier since you can address the square roots of the numerator and the denominator individually. In our original problem, \( \sqrt{\frac{11}{9}} \) is transformed into \( \frac{\sqrt{11}}{\sqrt{9}} \). This separation makes it clear how to proceed with further simplification of the expression.
Ensuring correct application of this property helps to maintain the mathematical equivalence and makes complex expressions much simpler to understand and solve.

Square Root of a Fraction

Understanding the square root of a fraction is essential when simplifying radical expressions. A fraction represents a division, and when such fractions are under a square root sign, they may initially appear complex. However, applying the Quotient Property of Square Roots can greatly ease this challenge.
When confronted with \( \sqrt{\frac{11}{9}} \), the expression can be split into separate square roots: \( \frac{\sqrt{11}}{\sqrt{9}} \). The important aspect to remember is to deal with the numerator and the denominator separately as independent problems:

• simplify \( \sqrt{11} \), which remains as it is due to 11 being a prime number
• simplify \( \sqrt{9} = 3 \), as 9 is a perfect square

This patient approach not only simplifies the process but also provides a clearer view of the steps involved in working with square roots of fractions.
Students often find this method helpful as it breaks down the complexities and provides a structured pathway to obtain the simplest form.

Prime Numbers in Radicals

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. In the context of simplifying square roots within radical expressions, an understanding of prime numbers is critical because a prime number under a square root cannot be simplified further.
For instance, in \( \sqrt{11} \), since 11 is a prime number, it cannot be broken down into smaller factors that would allow further simplification. This is an important final step while simplifying radical expressions. Even if the entire expression can be broken down or separated, the presence of a square root with a prime number indicates that part remains in its simplest possible form.

• Recognize that a prime number remains "as is" within a square root.
• No further reduction is possible unless more mathematical properties or external factors influence the expression.

Recognizing prime numbers in radicals ensures accuracy in simplification and prevents unnecessary steps.