Problem 100 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\sqrt{\frac{81}{121}}$$

Step-by-Step Solution

Answer

The value is \( \frac{9}{11} \).

Step 1: Simplify the Square Root of the Fraction

The given problem is to find \( \sqrt{\frac{81}{121}} \). The property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) allows us to take the square root of the numerator and the denominator separately. Thus, we have:\[ \sqrt{\frac{81}{121}} = \frac{\sqrt{81}}{\sqrt{121}} \]

Step 2: Calculate the Square Roots

Now, we calculate the square roots of the numerator and denominator:- \( \sqrt{81} = 9 \) because \( 9 \times 9 = 81 \).- \( \sqrt{121} = 11 \) because \( 11 \times 11 = 121 \).So, \( \frac{\sqrt{81}}{\sqrt{121}} = \frac{9}{11} \).

Step 3: Simplify the Fraction

The fraction \( \frac{9}{11} \) is already in its simplest form because 9 and 11 have no common factors other than 1. Therefore, the simplest form of the given expression is \( \frac{9}{11} \).

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. Simplifying a fraction means dividing both the numerator and the denominator by their greatest common divisor (GCD) until they are relatively prime (they have no common factors except 1). This makes the fraction easier to work with and understand.
For example, with the fraction \( \frac{81}{121} \), we cannot simplify it further because 81 and 121 have no common factors other than 1. This fraction is already in its simplest form. Simplifying fractions is essential in math because it allows us to work with the easiest version of a number. It also helps in comparing, adding, subtracting, multiplying, or dividing fractions.

Numerator and Denominator

Every fraction has a numerator and a denominator. These two components are crucial because they help express a part of a whole.
The numerator is the top part of a fraction and tells you how many parts of a whole you have. The denominator is the bottom part and shows how many equal parts the whole is divided into. Understanding their roles is essential for working with fractions.
For example, in the fraction \( \frac{9}{11} \), 9 is the numerator, indicating the number of parts taken, and 11 is the denominator, showing the total number of equal parts. Proper handling of numerators and denominators is key when performing operations with fractions, such as addition, subtraction, multiplication, and division.

Properties of Square Roots

Square roots are a fundamental concept in mathematics. The square root of a number \( a \) is a value that, when multiplied by itself, gives \( a \). The property that allows us to handle square roots in fractions is \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This means you can take the square root of each part of the fraction separately.
This property is incredibly useful for simplifying expressions involving square roots. It allows us to identify simpler values, making calculations more straightforward. In the problem \( \sqrt{\frac{81}{121}} \), applying this property, we separate the fraction into \( \frac{\sqrt{81}}{\sqrt{121}} \). Calculating further, we find \( \sqrt{81} = 9 \) and \( \sqrt{121} = 11 \), leading us to the simplified fraction \( \frac{9}{11} \).
Understanding the properties of square roots not only simplifies complex expressions but also aids in solving equations in algebra and geometry.