Problem 23 - Algebra 1

Question

Simplify the expression. $$\sqrt{\frac{11}{16}}$$

Step-by-Step Solution

Answer

\(\frac{\sqrt{11}}{4}\)

Step 1: Breaking The Square Root

A key property of square roots is that the square root of the division of two numbers is equal to the division of the square roots of these numbers. Knowing this property, we can simplify the given expression by taking the square root of both the numerator (11) and the denominator (16) separately. This will give us as \(\sqrt{\frac{11}{16}} = \frac{\sqrt{11}}{\sqrt{16}} \).

Step 2: Calculating the Square Root

Now, we continue by calculating the square root for both the numerator and the denominator. The square root of 11 cannot be simplified any further but the square root of 16 is 4. Therefore, the term \(\sqrt{16}\) becomes 4. This gives us \( \frac{\sqrt{11}}{4} \).

Key Concepts

Square Root Properties

When working with square roots, it's essential to understand how they relate to the fundamental properties of arithmetic operations. One of the key square root properties is the ability to split the square root across a fraction. In other words, the square root of a fraction like \(\frac{a}{b}\) is the same as the fraction comprised of the square roots of its numerator and denominator, \(\frac{\sqrt{a}}{\sqrt{b}}\). This property simplifies problems considerably, especially when dealing with perfect squares.

Another important property is that the square root of a product can be expressed as the product of the square roots of its factors: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Likewise, the square root of a quotient can be expressed as the quotient of the square roots, which you've seen in the exercise, \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). These properties allow for the simplification of radical expressions and are the cornerstone of working efficiently with radical numbers.

Rationalizing Denominators

The concept of rationalizing denominators involves modifying a fraction so that the denominator is a rational number rather than a radical. What's the point, you ask? Well, it's generally easier to work with and understand rational numbers than it is to perform operations with radicals. When you have a square root in the denominator, like \(\frac{1}{\sqrt{2}}\), you can rationalize it by multiplying both the numerator and the denominator by the square root that's in the denominator.

Why does this work? Because \(\sqrt{2} \times \sqrt{2}\) equals 2, turning our denominator into a rational number. This technique is vital when dealing with more complex algebraic expressions and is often required to present your final answer in a standardized form, which is generally more acceptable in mathematics.

Radical Expressions

Radical expressions include numbers under the radical sign, and dealing with them can often seem daunting. However, understanding their properties and how to manipulate them can make solving these expressions much more approachable. A radical expression is any mathematical expression containing a radical symbol (\(\sqrt{}\)) with an index, signifying the degree of the root.

When simplifying radical expressions, it's important to look for factors that are perfect powers of the index. For example, in simplifying \(\sqrt{18}\), we search for the largest perfect square factor, which is 9, and rewrite \(\sqrt{18}\) as \(\sqrt{9 \times 2}\), or \(3\sqrt{2}\). These simplifications lead to solving problems more effectively and pave the way for further operations such as addition, subtraction, or multiplication with radicals. Understanding radical expressions is crucial for any student's mathematical toolkit.